math-2

Math Tutor 1
CHAPTER-02
ev¯Íe msL¨v (Real Number)
01.01 A¼ (Digit)
 wnmvewbKvk I MYbvi Kv‡R e¨eüZ wPý ev cÖZxK|
 MwY‡Z †gvU 10 wU A¼ i‡q‡Q| †hgbt 0, 1, 2, 3, 4, 5, 6, 7, 8 Ges 9|
 ¯^v_©K A¼t 9 wU | †hgbt 1, 2, 3, 4, 5, 6, 7, 8 Ges 9|
 AfveÁvcK A¼ t 1 wU| 0 †K mvnvh¨Kvix msL¨v ejv nq| †hgbt 0|
01. ¯^v_©K A¼ KqwU? cÖv_wgK we`¨vjq cÖavb wkÿK-2005
5 wU 9 wU 7 wU 8 wU DËi: L
02. wb‡Pi †KvbwU ¯^v_©K A¼ bq? wbe©vnx Awdmvi, evwYR¨ gš¿Yvj‡qi Avg`vwb-ißvwb Awa`ßi-15
2 5 0 7 DËi: M
03. †Kvb msL¨v‡K mvnvh¨Kvix msL¨v ejv nq? cÖkvmwbK Kg©KZv© I cv‡m©vbvj Awdmvi, wbe©vPb Kwgkb-2004
0 1 2 5 DËi: K
01.02 msL¨v (Number)
 GK ev GKvwaK A¼ wg‡j msL¨v •Zwi nq Ges msL¨vi †k‡l wU, Uv, Lvbv _v‡K| †hgb- 5 wU Kjg, 13 Lvbv eB |
GLv‡b, 5 Ges 13 n‡”Q msL¨v|
 msL¨vq e¨eüZ A¼¸wji gvb(Value) `yÕfv‡e wPwýZ Kiv hvq| (i) ¯^Kxq gvb (Face Value) (ii) ¯’vbxq gvb
(Place Value)
01.03 01 †_‡K 100 ch©šÍ µwgK msL¨v I cÖ‡qvRbxq Z_¨vewj
1 11 21 31 41 51 61 71 81 91
2 12 22 32 42 52 62 72 82 92
3 13 23 33 43 53 63 73 83 93
4 14 24 34 44 54 64 74 84 94
5 15 25 35 45 55 65 75 85 95
6 16 26 36 46 56 66 76 86 96
7 17 27 37 47 57 67 77 87 97
8 18 28 38 48 58 68 78 88 98
9 19 29 39 49 59 69 79 89 99
10 20 30 40 50 60 70 80 90 100
¸iæZ¡c~Y© Z_¨vewjt nv‡Z Kj‡g †kLvi Rb¨ Dc‡iv³ Q‡Ki mv‡_ wb‡Pi Z_¨¸wj wgwj‡q wbb, Zvn‡j mn‡RB g‡b
_vK‡e|
 1 †_‡K 100 ch©šÍ wjL‡Z ev ¸Y‡Z 0 Av‡Q 11 wU|
 1 †_‡K 100 ch©šÍ wjL‡Z ev ¸Y‡Z 1 Av‡Q 21 wU|
 1 †_‡K 100 ch©šÍ wjL‡Z ev ¸Y‡Z 2, 3, 4, 5, 6, 7, 8, 9 Av‡Q 20wU K‡i|
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†ewmK, GgwmwKD I wjwLZ Av‡jvPbv
 

2Math Tutor
01. 1 †_‡K 100 ch©šÍ wjL‡Z Ô4Õ A¼wU KZevi Av‡m? SouthestBankLtdProbationaryOfficer:12
10 11 19 20 DËi: N
02. 1 †_‡K 100 ch©šÍ ¸Y‡Z, 5 msL¨vwU KZevi Av‡m? 28ZgwewmGm
10 11 28 19 DËi:
Tips: cÖ`Ë Ackb¸‡jv‡Z mwVZ DËi †bB| mwVK DËi n‡e 20|
03. 1 †_‡K 100 ch©šÍ ¸Y‡Z, 8 msL¨vwU KZevi Av‡m? BangladeshBank AssistantDirector:13
11 20 80 70 10 DËi: L
04. 1 †_‡K 100 ch©šÍ ¸Y‡Z, 9 msL¨vwU KZevi Av‡m? 28ZgwewmGm(gb¯ÍvwË¡K)
11 14 15 18 20 DËi: O
01.04 ÿz`ªZg I e„nËg msL¨v
cÖ`Ë A¼ e„nËg msL¨v ¶z`ªZg msL¨v
GK A¼ wewkó msL¨vi 9 1
`yB A¼ wewkó msL¨vi 99 10
wZb A¼ wewkó msL¨vi 999 100
Pvi A¼ wewkó msL¨vi 9999 1000
¯^vfvweK msL¨vi ÿz`ªZg m`m¨ n‡”Q 1|
01. ¯^vfvweK msL¨vi ÿz`ªZg m`m¨ †KvbwU? cÖavbgš¿xiKvh©vj‡qimnKvixcwiPvjK,M‡elYvKg©KZ©v,†Uwj‡dvbBwÄwbqviImnKvixKw¤úDUvi†cÖvMÖvgvi:13
1 0 Amxg me¸‡jv DËi: K
(K) mgvb msL¨K e„nËg msL¨v I ÿz`ªZg msL¨v Ô‡hvMÕ Kivt
9 9 9 9 99 9 999 9 9999
+1 + 10 + 100 + 1000 + 10000
10 10 9 10 99 10 999 10 9999
Dc‡iv³ QK †_‡K eySv hv‡”Q, mgvb msL¨K e„nËg I ÿz`ªZg msL¨vi †hvMdj PvB‡j Rv÷ cÖ_g 9 A¼wU 10 wj‡L
evKx †h KqwU 9 _vK‡e †m¸‡jv H Ae¯’vq †i‡L w`‡j †h msL¨vwU Avm‡e †mwUB DËi| †hgb Pvi A‡¼i e„nËg I
ÿz`ªZg msL¨vi †hvMdj n‡e- cÖ_g 9 A¼wU 10 G iƒcvšÍi Kiæb Ges evKx wZbwU ÔbqÕ (999) H Ae¯’vq †i‡L w`b,
Zvn‡j †hvMdj `uvov‡”Q 10999| A_©vr, †hvMdj n‡e- GKwU Ô`kÕ Ges evKx wZbwU ÔbqÕ|
02. cvuP A‡¼i e„nËg I ¶y`ªZg msL¨vi †hvMdj KZ? ivóªvqË e¨vsK wmwbqi Awdmvi : 00
109999 89999 100009 †KvbwUB bq DËi: K
(L) mgvb msL¨K e„nËg msL¨v I ÿz`ªZg msL¨v Ôwe‡qvMÕ Kivt
9 9 9 9 99 9 999 9 9999
- 1 - 10 - 100 - 1000 - 10000
8 8 9 8 99 8 999 8 9999
Dc‡iv³ QK †_‡K eySv hv‡”Q, mgvb msL¨K e„nËg I ÿz`ªZg msL¨vi we‡qvMdj PvB‡j Rv÷ cÖ_g 9 A¼wU 8 wj‡L
evKx †h KqwU 9 A¼ _vK‡e †m¸‡jv H Ae¯’vq †i‡L w`‡j †h msL¨vwU Avm‡e †mwUB DËi| †hgb Pvi A‡¼i e„nËg
I ÿz`ªZg msL¨vi we‡qvMdj n‡e- cÖ_g 9 A¼wU 8 G iƒcvšÍi Kiæb Ges evKx wZbwU ÔbqÕ (999) H Ae¯’vq †i‡L
w`b, Zvn‡j we‡qvMdj `uvov‡”Q 8999| A_©vr, we‡qvMdj n‡e- 1wU Ô8Õ Ges evKx 3wU Ô9Õ|
03. Pvi A‡¼i e„nËg I ¶y`ªZg msL¨vi cv_©K¨ KZ? ivóªvqË e¨vsK Awdmvi t 97
10999 8999 1009 1999 DËi: L
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Math Tutor 3
04. 6 A‡¼i e„nËg I ¶z`ªZg msL¨vi cv_©K¨ KZ? †mvbvwj, RbZv I AMÖYx e¨vsK wmwbqi Awdmvi : 08
888889 899999 988888 999888 DËi: L
(M) ÿz`ªZg msL¨v †_‡K e„nËg msL¨v we‡qvM Kivi mgq ÿz`ªZg msL¨v A‡cÿv e„nËg msL¨vi GKwU ÔwWwRU/A¼Õ Kg
n‡j we‡qvMdj memgq 1 nq|
10 100 1000 10000 100000
- 9 - 99 - 999 - 9999 - 99999
1 1 1 1 1
05. cvuP A‡¼i ¶z`ªZg msL¨v I Pvi A‡¼i e„nËg msL¨vi AšÍi KZ? 29Zg wewmGm
9 10 1 -1 DËi: M
(N) weweat
06. `yBkZ bq nvRvi ‡PŠÏ Ges wZivbeŸB nvRvi mvZ kZ bq Gi AšÍi KZ? weweG 92-93
116315 115315 116305 115305 DËi: N
209014 - 93709 = 115305|
07. 0, 1, 2, Ges 3 Øviv MwVZ Pvi A‡¼i e„nËg Ges ÿz`ªZg msL¨vi we‡qvMdj- cjøxwe`y¨Zvqb†ev‡W©imn.mwPe/cwiPvjK2017
3147 2287 2987 2187
0, 1, 2, 3 Øviv Pvi A‡¼i e„nËg msL¨v = 3210 Ges ÿz`ªZg msL¨v= 1023 | myZivs, msL¨v `ywUi we‡qvMdj
= 3210 - 1023 = 2187| DËi: N
01.05 e‡M©i AšÍi
 †R‡b wbb – 01
 `ywU µwgK msL¨vi eM©‡K we‡qvM Ki‡j †h msL¨v cvIqv hvq, H µwgK msL¨v `ywU‡K †hvM Ki‡jI GKB msL¨v cvIqv
hvq| †hgb-2 I 3 Gi †hvMdj Ges Zv‡ì e‡M©i Aš‘i mgvb| A_©vr, 2 + 3 = 5  2
3 - 2
2 = 9 - 4 = 5|
GKBfv‡e, 3 I 4 Gi †hvMdj Ges Zv‡ì e‡M©i AšÍi mgvb| A_©vr, 3 + 4 = 7  2
4 - 2
3 = 16 - 9 = 7
 kU©KvU †UKwbK: ÿz`ªZg msL¨v wbY©q =
2
1
-
AšÍi
i
M©
e‡
 e„nËg msL¨v wbY©q =
2
1
AšÍi
i
M©
e‡ 
01. `ywU µwgK c~Y© msL¨vi e‡M©i AšÍi 47| eo msL¨vwU KZ? 26Zg wewmGm (wkÿv); wdb¨vÝ wgwbw÷ª -2009
24 25 26 30
†h‡nZz `ywU µwgK msL¨vi e‡M©i AšÍi 47 †`qv Av‡Q, †m‡nZz Avgiv ej‡Z cvwi 47 n‡”Q `ywU µwgK msL¨vi
†hvMdj| 47 Gi gv‡S `ywU µwgK msL¨v 23 I 24 Av‡Q, hv‡ì gv‡S eo msL¨vwU n‡”Q 24| DËi: K
 kU© †UKwb‡K mgvavb: eo msL¨vwU =
2
1
AšÍi
i
M©
e‡ 
=
2
1
47 
= 24|
02. `ywU µwgK c~Y©msL¨v wbY©q Kiæb, hv‡ì e‡M©i AšÍi 93|wgwbw÷ª Ae I‡gb - 2007
46, 47 44, 45 43, 44 50, 51 DËi: K
Ackb ¸‡jvi gv‡S †h `ywU µwgK msL¨vi †hvMdj 93 †mwUB DËi| GLv‡b Ackb †Z 46 + 47 = 93|
03. `ywU µwgK c~Y© msL¨vi e‡M©i AšÍi 63| †QvU msL¨vwU KZ?
30 31 32 33
63 Gi gv‡S 2wU µwgK 31 I 32 Av‡Q, hv‡ì gv‡S †QvU msL¨vwU n‡”Q 31|
mgvavb
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mgvavb
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4Math Tutor
 kU© †UKwb‡K mgvavbt
2
1
-
63
=
2
62
= 31 | myZivs, †QvU msL¨vwU 31| DËit L
 †R‡b wbb - 02
 `ywU msL¨v ej‡Z †h‡Kvb 2wU msL¨v‡K eySvq, ZvB x I y a‡i A¼ Kiv nq|
 `ywU µwgK msL¨v ej‡Z ci ci `ywU msL¨v‡K eySvq, hv‡`i cv_©K¨ memgq 1 _v‡K, ZvB GKwU msL¨v x I AciwU
x + 1 a‡i A¼ Kiv nq| cÖ_gwU‡K ejv nq †QvU msL¨v Ges c‡iiwU‡K ejv nq eo msL¨v|
 †Kvb mgm¨vi †ÿ‡Î ARvbv ivwk/msL¨v a‡i mgvav‡bi DËg Dcvq n‡”Q ÔkZ©g‡Z/cÖkœg‡ZÕ mgxKiY `vo Kiv‡bv|
 wjwLZ mgvavb Kivi wbqgt
G ai‡Yi A‡¼ `ywU µwgK msL¨vi Ôe‡M©i AšÍiÕ †`qv _vK‡jI ÔµwgK msL¨vÕ `ywU †`qv _v‡K bv| GRb¨ G ai‡Yi
mgm¨v mgvav‡bi Rb¨ ïiæ‡ZB µwgK msL¨v `ywU a‡i wb‡q mgvavb Ki‡Z nq| wb‡Pi mgm¨vwU †`Lyb-
04. `yBwU µwgK c~Y© msL¨vi e‡M©i AšÍi 199| eo msL¨vwU KZ? 22Zg wewmGm; cÖvK-cÖv_wgK mnKvix wkÿK-2015
70 80 90 100 DËi: N
(cÖ`Ë mgm¨vwU‡Z `ywU µwgK msL¨vi e‡M©i AšÍi 199 †`qv Av‡Q| wKš‘ µwgK msL¨v `ywU †`qv bvB| GRb¨
ïiæ‡ZB msL¨v `ywU a‡i wb‡Z n‡e|)
g‡bKwi, †QvU msL¨vwU = x Ges eo msL¨vwU = x +1 (GKevi Ô†R‡b wbb-02Õ c‡o wbb)
(GLb cÖ`Ë mgm¨vi Av‡jv‡K GKwU ÔcÖkœg‡Z/kZ©g‡ZÕ mgxKiY `vo Kiv‡Z n‡e| GRb¨ `v‡M hv hv ejv n‡q‡Q, ZvB
Kiæb| `v‡M hv hv ejv n‡q‡Q- (1) µwgK msL¨v `ywU eM© n‡e, ZvB eM© K‡i †djyb- x2
Ges (x+1)2
| (2) eM© `ywUi
AšÍi n‡e, ZvB Gevi we‡qvM K‡i †djyb- (x+1)2
- x2
. (3) †k‡l ejv n‡q‡Q, GB e‡M©i AšÍi mgvb n‡”Q 199|
ZvB AvcwbI †mfv‡e wj‡L †djyb- (x+1)2
- x2
= 199| e¨m&, Gfv‡eB `vwo‡q †Mj ÔcÖkœg‡ZÕ mgxKiYwU!!)
cÖkœg‡Z, (x+1)2
- x2
= 199
ev, x2
+ 2x +1 - x2
= 199 [(a+b)2
= a2
+2ab +b2
Abymv‡i]
ev, 2x = 199 - 1
x =
2
1
-
199
=
2
198
= 99
AZGe, eo msL¨vwU =x +1 = 99 + 1 = 100|
05. `yBwU µwgK ¯^vfvweK msL¨vi e‡M©i AšÍi 151 n‡j msL¨v `yBwU KZ? WvKI†Uwj‡hvMv‡hvMwefv‡MiAaxbWvKAwa`ß‡iiwewìs
Ifviwkqvi2018
46, 47 75, 76 67, 68 54, 55 DËi: L
g‡b Kwi, †QvU msL¨vwU = x Ges eo msL¨vwU = x+1
cÖkœg‡Z, (x + 1)2
- x2
= 151 ev, x2
+ 2x + 1 - x2
= 151 ev, 2x = 151 - 1 ev, x =
2
150
= 75
†QvU msL¨vwU = 75 Ges eo msL¨vwU = 75 + 1 = 76
PP©v Kiæb
06. `ywU µwgK ¯^vfvweK msL¨vi e‡M©i AšÍi 45 n‡j, msL¨v `ywU - mvaviYcy‡jiAvIZvqwewfbœ gš¿Yvj‡qimnKvix†cÖvMÖvgviDcmnKvixcÖ‡KŠkjx,
cÖkvmwbKKg©KZ©vIe¨w³MZKg©KZ©v:16
21, 22 22, 23 23, 24 20, 21 DËi: L
07. `ywU µwgK msL¨vi e‡M©i AšÍi 37| msL¨v `yBwU wK wK? evsjv‡`k†ijI‡qDcmnKvixcÖ‡KŠkjx(wmwfj):16
12, 13 15, 16 18, 19 20, 21 DËi: M
08. `ywU µwgK msL¨vi e‡M©i AšÍi 25| GKwU msL¨v 12 n‡j, Aci msL¨vwU - ¯^v¯’¨gš¿Yvj‡qiDcmnKvixcÖ‡KŠkjx(wmwfj):16
5 9 11 13 DËi: N
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mgvavb
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mgvavb
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Math Tutor 5
09. `ywU µwgK msL¨vi e‡M©i AšÍi 11 n‡j, msL¨v `yBwUi e‡M©i mgwó KZ? gwnjvIwkïwelqKgš¿Yvj‡qiAaxbgwnjvwelqKKg©KZ©v:16
16 17 61 71 DËi: M
10. `ywU µwgK c~Y©msL¨v wbY©q Kiæb hv‡ì e‡M©i AšÍi 9 n‡e? Rbkw³,Kg©ms¯’vbIcÖwkÿYey¨‡iviBÝ÷ªv±i:18
4 Ges 5 5 Ges 6 6 Ges 7 7 Ges 8 DËi: K
11. wb‡Pi †Kvb µwgK c~Y© msL¨vØ‡qi e‡M©i AšÍi 43? L¨v`Äwa`ß‡iiLv`¨cwi`k©K/Dc-Lv`¨cwi`k©K:11
21 Ges 22 22 Ges 23 23 Ges 24 24 Ges 25 DËi: K
12. `yBwU µwgK ALÐ msL¨vi e‡M©i AšÍi 49 n‡j, †QvU msL¨vwU n‡e- wewfbœ gš¿Yvjq/wefvM/Awa`ß‡iie¨w³MZ
Kg©KZv© (mvaviY)2018
19 20 24 25 DËi: M
13. ci ci `ywU c~Y© msL¨v wbY©q Ki hv‡ì e‡M©i cv_©K¨ n‡e 53-Rbkw³,Kg©ms¯’vbIcÖwkÿYey¨‡iviDcmnKvixcwiPvjKt01
25 Ges 26 26 Ges 27 27 Ges 28 28 Ges 29 DËi: L
14. `ywU µwgK c~Y©msL¨vi e‡M©i AšÍi 79 n‡j eo msL¨vwU KZ?gv`K`ªe¨wbqš¿YAwa`ß‡iimnKvixcwiPvjK-2013
40 35 45 100 DËi: K
15. `ywU µwgK c~Y©msL¨vi e‡M©i AšÍi 111 n‡j eo msL¨vwU KZ?ciivóª gš¿Yvj‡qie¨w³MZKg©KZv©-2006
54 55 56 57 DËi: M
16. `ywU µwgK c~Y©msL¨vi e‡M©i AšÍi 197| msL¨vØq KZ?†bŠcwienbgš¿YvjqIcÖwZiÿvgš¿Yvj‡qicÖkvmwbKKg©KZv©-2013
97, 98 96, 97 98, 99 99, 100 DËi: M
01.06 hZ ZZ
 †R‡b wbb -03
Dc‡ii QKwU jÿ¨ Kiæb, QKwU‡Z †`Lv hv‡”Q,20 msL¨vwU 15 †_‡K 5 †ewk GKBfv‡e 20 msL¨vwU 25 †_‡K 5 Kg|
Gevi GKwU cÖkœ ùvo Kiv‡bv hvK-GKwU msL¨v 15 †_‡K hZ eo 25 †_‡K ZZ †QvU| msL¨vwU KZ?QKvbymv‡i, msL¨vwU
n‡”Q 20| gRvi e¨vcvi n‡jv, 15 I 25msL¨v `ywU †hvM K‡i 2 Øviv fvM Ki‡jB 20 cvIqv hvq| A_©vr, †Kvb cÖ‡kœ
ÒGKwU msL¨v --- †_‡K hZ eo ---- †_‡K ZZ †QvU| msL¨vwU KZ?Ó Giƒc ejv _vK‡j Avcwb mivmwi cÖ‡kœ cÖ`Ë msL¨v
`ywUi Mo Ki‡jB msL¨vwU cvIqv hv‡e| myZivs, msL¨vwU =
2
25
15 
= 20|
01. GKwU msL¨v 650 n‡Z hZ eo 820 †_‡K ZZ †QvU| msL¨vwU KZ? 22Zg wewmGm
730 735 800 780 DËi: L
msL¨v `ywU 650 I 820 †hvM Kiæb Ges 2 Øviv fvM Kiæb, DËi n‡e 735|
02. GKwU msL¨v 553 n‡Z hZ eo 651 †_‡K ZZ †QvU| msL¨vwU KZ? [mve †iwR÷ªvi 1992]
603 601 605 602 DËi: N
msL¨v `ywU 553 I 651†hvM Kiæb Ges 2 Øviv fvM Kiæb, DËi n‡e 602|
03. GKwU msL¨v 742 n‡Z hZ eo 830 †_‡K ZZ †QvU| msL¨vwU KZ? [_vbv I †Rjv mgvR‡mev Awdmvi 1999]
780 782 790 786 DËi: N
msL¨v `ywU 742 I 830 †hvM Kiæb Ges 2 Øviv fvM Kiæb, DËi n‡e 786|
 wjwLZ mgvavb Kivi wbqgt
04. GKwU msL¨v 301 †_‡K hZ eo 381 †_‡K ZZ †QvU| msL¨vwU KZ? [30Zg wewmGm]
340 341 342 344 DËi t L
(mgm¨vwU‡Z GKwU msL¨vi K_v ejv n‡”Q hv 310 †_‡K hZUzKz eo n‡e, wVK 381 †_‡K ZZUzKzB †QvU n‡e|
mgvavb
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mgvavb
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mgvavb
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mgvavb
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+ 5 = 20 25
15 + 5 =

6Math Tutor
Gevi Avcbv‡K ej‡Z n‡e msL¨vwU KZ? ejyb‡Zv msL¨vwU KZ †mUv Avcwb Rv‡bb? bv Rv‡bb bv| Zvi gv‡b GwU
GKwU ARvbv ivwk| ZvB Avcbv‡K ïiæ‡ZB GKwU msL¨v x a‡i wb‡q A¼ Klv ïiæ Ki‡Z n‡e| )
g‡bKwi, msL¨vwU = x
(Gevi `vMwUi `y‡qKevi co–b Ges wb‡Pi QKwU †`‡L mgm¨vwU wfZi †_‡K eySvi †Póv Kiæb -
cÖ_‡g eySvi †Póv Kiæb- ejv n‡q‡Q GKwU msL¨v (x)301 †_‡K hZUyKz eo n‡e A_©vr, x †_‡K 301 we‡qvM Ki‡j
†h gvb ‡ei n‡e , 381 †_‡K H GKwU msL¨v (x) we‡qvM Ki‡j †h gvb †ei n‡e Zvi mgvb| GLb Avgv‡ì x Gi
gvb †ei K‡i welqwU cÖgvY Kiv Riæwi| GRb¨ cÖkœvbymv‡i Pjyb GKwU kZ© ùvo Kiv‡bv hvK|)
kZ©g‡Z, x - 301 = 381 - xev, x +x = 381 + 301 ev, 2x = 381 + 301 ev, x =
2
301
381
ev, x =
2
682
= 341 (DËi)
 m¤ú~Y© mgm¨vwU wK¬qviwj eySvi Rb¨ Dc‡ii QKwU bZzb K‡i †`Lyb|
 civgk©: cÖwZwU A¼ evievi we¯ÍvwiZ Kiæb, †`L‡eb hLb wei³ jvM‡Q ZLb g‡bi ARv‡šÍB kU©‡UKwbK •Zwi n‡q †M‡Q!
05. GKwU msL¨v 560 †_‡K hZ Kg, 380 †_‡K Zvi mv‡o wZb¸Y †ewk| msL¨vwU KZ? Dc‡Rjv_vbvwkÿvAwdmvi(AETO):10
450 470 520 500 DËi: M
mgm¨vwU‡Z †h msL¨vwU †ei Ki‡Z ejv n‡q‡Q- †mB ÔmsL¨vwUÕ I Ô560ÕGi gv‡S hZUzKz eëavb Ges †mB
ÔmsL¨vwUÕ I Ô360ÕGi gv‡S hZUzKz eëavb , Zv hw` Avgiv Zzjbv Kwi Zvn‡j `ywU eëav‡bi cv_©K¨ n‡e mv‡o
wZb¸Y †ewk n‡e| QKwU jÿ¨ Kiæb, welqwU wK¬qvi n‡q hv‡e|
Gevi Ackb †_‡K ÔmsL¨vwUÕi gvb ewm‡q Df‡qi cv‡k¦©i eëavb wbY©q Kiæb Ges Dfq eëavb Zzjbv Kiæb †mwU
GKwU Av‡iKwUi mv‡o wZb¸Y wKbv?
(GLv‡b 110 Gi mv‡o wZb¸Y 70 n‡e bv) (GLv‡b 90 Gi mv‡o wZb¸Y 90 n‡e bv)
(GLv‡b 40 Gi mv‡o wZb¸Y 140|) mwVK DËi (GLv‡b 60 Gi mv‡o wZb¸Y 120 n‡e bv)
 civgk©t cixÿvi LvZvq kU©Kv‡U we‡qvM K‡i wb‡eb|
N
M
L
K
mgvavb
N
M
L
K
x 301 381 x
cv_©K¨ (x - 301) = cv_©K¨ (381 -x)
341 301 381 341
341 - 301 = 40 eo = 381 - 341 = 40 †QvU
Gevi `vMwU co–b †Zv ey‡Sb wKbv?
GKwU msL¨v (341) 301 †_‡K hZ (40) eo
381 †_‡K ZZ (40) †QvU|
#ey‡S ey‡S mgvavb Kiæb, MwYZ fq `~i Kiæb
msL¨vwU 560
380
eëavb eëavb
mv‡o wZb¸Y †ewk
470 560
380
450 560
380
eëavb
450-380=70
eëavb
560 - 450 =110
eëavb
470-380=90
eëavb
560 - 470 =90
520 560
380 500 560
380
eëavb
500 - 380=120
eëavb
560 - 500 = 60
eëavb
520 - 380=140
eëavb
560 - 520 = 40

Math Tutor 7
06. 765 †_‡K 656 hZ Kg, †Kvb msL¨vi 825 †_‡K ZZUzKz †ewk? ¯îvóª gš¿YvjqewnivMgbIcvm‡cvU© Awa:mn:cwiPvjK:11;kÖgAwa:kÖg
Kg©KZ©vGesRbmsL¨vIcwieviKj¨vYKg©KZ©v:03
932 933 934 935 DËi: M
cÖ`Ë cÖ‡kœ ejv n‡q‡Q, 765 I 656 Gi gv‡S hZUzKz eëavb, msL¨vwU I 825 Gi gv‡S ZZUzKzB eëavb|
myZivs, msL¨vwU = 109 + 825 = 934|
PP©v Kiæb
07. GKwU msL¨v 31 †_‡K hZ †ewk, 55 †_‡K ZZ Kg, msL¨vwU KZ? cÖwZiÿvgš¿Yvj‡qiwmwfwjqvb÷vdAwdmviGes
mnKvixcwiPvjK2016;ciivóª gš¿Yvj‡qimvBdviAwdmvi:12
39 41 43 45 DËi: M
08. GKwU msL¨v 999 †_‡K hZ †QvU 797 †_‡K ZZ eo| msL¨vwU KZ? Lv`Äwa`ß‡iiLv`¨/Dc-Lv`¨cwi`k©K-2011
897 898 899 900 DËi: L
09. GKwU msL¨v 742 n‡Z hZ eo 830 n‡Z ZZ †QvU, msL¨vwU KZ? mgvR‡mevAwa`ß‡iiBDwbqbmgvRKg©xwb‡qvMcixÿv2016
780 782 790 786 DËi: N
10. GKwU msL¨v 470 †_‡K hZ eo 720 †_‡K ZZ †QvU| msL¨vwU KZ? gnv-wnmvewbixÿKIwbqš¿‡KiKvh©vj‡qAwWUi:15
565 595 615 †Kv‡bvwUB bq DËi: L
11. GKwU msL¨v 100 †_‡K hZ eo 320 †_‡K ZZ †QvU| msL¨vwU KZ? cjøxDbœqbImgevqwefv‡MiGKwUevwoGKwULvgvicÖK‡íi
Dc‡Rjvmgš^qKvix:17;K…wlm¤úªmviYAwa:mnKvixK…wlKg©KZ©v:16
120 210 220 †Kv‡bvwUB bq DËi: L
01.07 µwgK msL¨vi ¸Ydj
 †R‡b wbb -04
 µwgK msL¨v : x x + 1 x + 2 x + 3 µwgK msL¨v (Gfv‡e a‡i wb‡eb)
1 1 + 1 1 + 2 1 + 3 Dc‡iv³ µwgK msL¨v¸‡jv‡Z x =1 emv‡j 1, 2, 3, 4
1 2 3 4 BZ¨vw` µwgK msL¨v¸‡jv †c‡q hv‡eb|
 µwgK †Rvo: x x + 2 x + 4 x + 6 µwgK †Rvo msL¨v (Gfv‡e a‡i wb‡eb)
2 2 + 2 2 + 4 2 + 6 Dc‡iv³ µwgK †Rvo msL¨v¸‡jv‡Z x = 2 emv‡j 2, 4,
2 4 6 8 6, 8 BZ¨vw` µwgK †Rvo msL¨v¸‡jv †c‡q hv‡eb|
 µwgK we‡Rvo: x x + 2 x + 4 x + 6 µwgK we‡Rvo msL¨v (Gfv‡e a‡i wb‡eb)
1 1 + 2 1 + 4 1 + 6 Dc‡iv³ µwgK we‡Rvo msL¨v¸‡jv‡Z x =1 emv‡j 1, 3,
1 3 5 7 5, 7 BZ¨vw` µwgK we‡Rvo msL¨v¸‡jv †c‡q hv‡eb|
 µwgK †Rvo I µwgK we‡Rvo Dfq‡ÿ‡Î x, x +2, x + 4, x + 6 GKBiKg †`‡L KbwdDRW n‡eb bv, KviY
GwU wbf©i K‡i x Gi gv‡bi Dci| x Gi gvb †Rvo wb‡j x, x + 2 … BZ¨vw` †Rvo µwgK msL¨v n‡e Ges x Gi
gvb we‡Rvo wb‡j x, x + 2 … BZ¨vw` we‡Rvo µwgK msL¨v n‡e|
 civgk©: G RvZxq mgm¨vmn MwY‡Zi †h‡Kvb As‡k fv‡jv Kivi Rb¨ 1 †_‡K 25 ch©šÍ bvgZv Aek¨B Rvb‡Z n‡e|
(K) `ywU µwgK msL¨vi ¸Ydj
01. `ywU µwgK abvZ¥K we‡Rvo msL¨vi ¸Ydj 255 n‡j msL¨vØq KZ? AgraniBankLtd.SeniorOfficer:13(cancelled)
N
M
L
K
N
M
L
K
N
M
L
K
N
M
L
K
N
M
L
K
mgvavb
N
M
L
K
825
765
656 msL¨vwU
eëavb
765 - 656 = 109
eëavb
msL¨vwU - 825 = 109

8Math Tutor
11, 13 13, 15 13, 17 15, 17 DËi: N
g‡bKwi, µwgK abvZ¥K we‡Rvo msL¨vØq = x, x + 2
(a‡i †bqv µwgK we‡Rvo msL¨v `ywUi ¸Ydj n‡e 255 Gi mgvb)
kZ©g‡Z, x (x + 2) = 255 ev, x2
+ 2x – 255 = 0 ev, x2
+ 17x – 15x – 255 = 0
ev, x(x + 17) – 15(x + 17) = 0 ev, (x + 17) (x – 15) = 0
x + 17 = 0 A_ev x – 15 = 0
∴ x = – 17 A_ev x = 15 (x Gi FYvZ¥K gvb MÖnY‡hvM¨ bq)
AZGe, µwgK abvZ¥K we‡Rvo msL¨vØq x = 15 I x + 2 = 17 |
 Ackb †_‡K kU©KvUt Ackb¸‡jvi msL¨vØq ¸Y K‡i hvi ¸Ydj 255 nq †mwUB DËi A_©vr, 15  17 = 255|
02. `ywU µwgK FYvZ¥K †Rvo c~Y©msL¨vi ¸Ydj 24 nq, Z‡e eo msL¨vwU KZ? IslamiBankLtd.ProbationaryOfficer:17
- 4 - 6 4 6 DËi: K
24 = (  4)(  6)| (  4) I (  6) Gi gv‡S eo msL¨vwU n‡”Q  4|
03. `ywU msL¨vi ¸Ydj 162| hw` e„nËg msL¨vwU ÿz`ªZg msL¨vi wØ¸Y nq, Z‡e e„nËg msL¨vwU KZ? BangladeshKrishi
BankLtd.SeniorOfficer:11
18 15 9 21 DËi: K
Ackb Gi 18 †K hw` e„nËg msL¨v wn‡m‡e a‡i †bqv nq, Zvn‡j ÿz`ªZg msL¨vwU n‡e 9|
18 I 9 Gi ¸Ydj n‡e 162 Ges e„nËg msL¨vwU ÿz`ªZg msL¨vi wØ¸Y|
(L) wZbwU µwgK msL¨vi ¸Ydj
04. wZbwU µwgK msL¨vi ¸Ydj 60 n‡j Zv‡`i †hvMdj KZ n‡e? ¯^ivóª gš¿Yvj‡qigv`K`ªe¨wbqš¿YAwa`ß‡iiDc-cwi`k©K:13;cwievi
cwiKíbvAwa`ß‡iimnKvixcwiKíbvKg©KZ©v:12;RvZxqivR¯^ †ev‡W©imnKvixivR¯^ Kg©KZ©v:12
20 12 15 14 DËi: L
we¯ÍvwiZ wbqgt g‡bKwi, msL¨v wZbwU = x, x + 1, x + 2
kZ©g‡Z, x(x+1) (x+2) = 60
ev, x(x2
+ 3x + 2) – 60 = 0
ev, x3
+ 3x2
+ 2x – 60 = 0
ev, x3
– 3x2
+ 6x2
– 18x + 20x – 60 = 0 ev, x2
(x–3) + 6x (x–3) + 20(x–3) = 0
ev, (x – 3) (x2
+ 6x + 20) = 0 GLv‡b, x – 3 = 0 ∴ x = 3
µwgK msL¨v wZbwU = 3, 4 I 5 |
myZivs, msL¨v wZbwUi †hvMdj = 3 + 4 + 5 = 12 |
 Drcv`‡K we‡køl‡Yi gva¨‡gt G RvZxq mgm¨v mgvav‡bi Rb¨ GB c×wZwU cvi‡d±|
(cÖ_‡g cÖ`Ë msL¨vwU‡K Drcv`‡K we‡kølY K‡i wb‡eb)
2 60 ∴ 60 = 2235 (Gevi GB Drcv`K¸‡jv †_‡K 3 wU µwgK msL¨v •Zwi Ki‡eb)
2 30 = 345
3 15 µwgK msL¨v wZbwUi †hvMdj = 3 + 4 + 5 = 12|
5
05. 3wU µwgK c~Y©msL¨vi ¸Ydj 120| msL¨v 3wUi †hvMdj KZ? 29Zg I 32Zg wewmGm
12 13 14 15 DËi: N
120 = 22235 = 456 myZivs, msL¨v wZbwUi †hvMdj = 4 + 5 + 6 = 15|
06. 3wU µwgK c~Y©msL¨vi ¸Ydj 210| msL¨v 3wUi †hvMdj KZ? evsjv‡`k e¨vsK Gwm÷¨v›U wW‡i±i -04, cÖv_wgK
mnKvix wkÿK 2010 (wZ¯Ív)]
mgvavb
N
M
L
K
mgvavb
N
M
L
K
K
mgvavb
N
M
L
K
mgvavb
N
M
L
K
mgvavb
N
M
L
K
awi, f(x) = x3
+ 3x2
+ 2x – 60 ∴ f(3) = 33
+ 3 . 32
+ 23
– 60 = 27 + 27 + 6 – 60
= 60 – 60 =0| †h‡nZz x = 3 emv‡j f(x) = 0 nq, †m‡nZz x – 3, f(x) Gi GKwU Drcv`K|

Math Tutor 9
12 14 15 18 DËi: N
210 = 2 35 7 = 56 7| myZivs msL¨v wZbwUi †hvMdj = 5 + 6 + 7 = 18|
07. 3wU µwgK c~Y©msL¨vi ¸Ydj 210| †QvU `ywU msL¨vi †hvMdj KZ? DBBL Assistant officer -09
5 11 20 13 DËi: L
210 = 235 7 = 56 7 | myZivs †QvU `ywU msL¨vi †hvMdj = 5 + 6 = 11|
08. wZbwU wfbœ c~Y©msL¨vi ¸Ydj 6| msL¨vÎ‡qi mgwói wØ¸‡Yi gvb KZ? IBA(MBA):88-89
12 4 18 36 DËi: K
6 = 123| msL¨vÎ‡qi mgwó = 1+2+3 = 6| AZGe, msL¨vÎ‡qi mgwói wØ¸Y = 62 = 12|
(6 Ggb GKwU msL¨v hvi Drcv`KÎ‡qi †hvMdj I ¸Ydj GKB n‡q _v‡K)
(M) cici/ µwgK wZbwU †Rvo ev we‡Rvo msL¨vi ¸Ydj
09. cici wZbwU †Rvo msL¨vi ¸Ydj 192 n‡j, Zv‡`i †hvMdj KZ? wkÿvgš¿Yvj‡qiRywbqiBÝUªv±i (†UK):16
10 18 22 24 DËi: L
192 = 2222223 = 468| ∴ †Rvo msL¨v wZbwUi †hvMdj = 4 + 6 + 8 = 18|
10. wZbwU wfbœ we‡Rvo msL¨vi ¸Ydj 15| ÿz`ªZg msL¨vwU KZ? IBA(MBA):88-89
12 4 18 None DËi: N
15 = 135 | ∴ ÿz`ªZg msL¨vwU = 1|
11. wZbwU µwgK †Rvo c~Y©msL¨vi ÿz`ªZg msL¨vwU e„nËgwUi wZb¸Y A‡cÿv 40 Kg| e„nËg msL¨vwU KZ? PÆMÖvg e›`‡ii
wb‡qvM 2017
14 17 18 19 DËi: M
g‡bKwi, µwgK msL¨v wZbwU x, x + 2, x + 4
(cÖkœvbymv‡i e„nËg msL¨vwU‡K wZb¸Y Ki‡j cÖvß ¸Ydj I ÿz`ªZg msL¨vi cv_©K¨ 40 n‡e, ZvB kZ©g‡Z e„nËg msL¨vwUi
wZb¸Y †_‡K ÿz`ªZg msL¨vwU we‡qvM K‡i mgvb mgvb 40 wjLyb)
kZ©g‡Z, 3(x + 4) – x = 40 ev, 3x + 12 – x = 40 ev, 2x = 40 – 12 ev, 2x = 28 ∴ x = 14
myZivs, wb‡Y©q e„nËg msL¨v = x + 4 = 14 + 4 = 18 |
01.08 `ywU msL¨vi †hvMdj, we‡qvMdj I ¸Ydj
01. `ywU msL¨vi mgwó 146 Ges AšÍi 18| msL¨vØq KZ? Agrani BankLtd.Officer(cash):13
74, 62 82, 64 84, 60 80, 62 DËi: L
g‡bKwi, eo msL¨vwU = x I †QvU msL¨vwU = y
x + y = 146 ---- (1) x  y = 18 ---- (2)
1 I 2 bs mgxKiY †hvM K‡i cvB,
x + y = 146
x  y = 18
2x = 164 ( 146 I 18 Gi †hvMdj)
 x =
2
164
= 82 (†hvMdj 2
 )
(1) †_‡K (2) bs mgxKiY we‡qvM K‡i cvB,
x + y = 146
x  y = 18
2y = 128 (146 I 18 Gi we‡qvMdj)
 y =
2
128
= 64 (we‡qvMdj  2)
mgvavb
N
M
L
K
mgvavb
N
M
L
K
mgvavb
N
M
L
K
mgvavb
N
M
L
K
mgvavb
N
M
L
K
mgvavb
N
M
L
K
mgvavb
N
M
L
K
 `ywU msL¨vi †hvMdj I we‡qvMdj †`qv _vK‡j eo
msL¨v (x) wbY©‡qi wbqg-
x =
2
AšÍi
qi
msL¨vØ‡
mgwó
qi
msL¨vØ‡ 
 `ywU msL¨vi †hvMdj I we‡qvMdj †`qv _vK‡j †QvU
msL¨v (y) wbY©‡qi wbqg-
y =
2
AšÍi
qi
msL¨vØ‡
-
mgwó
qi
msL¨vØ‡

10Math Tutor
AZGe, msL¨vØq 82 I 64|
02. `ywU msL¨vi †hvMdj 15 Ges we‡qvMdj 13| †QvU msL¨vwU KZ? ivóªvqË¡ e¨vsKAwdmvi:97
1 2 14 18 DËi: K
†hvMdj †_‡K we‡qvMdj we‡qvM K‡i 2 Øviv fvM Ki‡j †QvU msL¨vwU cvIqv hv‡e- (15  13)  2 = 1|
 civgk©t eo msL¨v PvB‡j †hvM Ges †QvU msL¨v PvB‡j we‡qvM K‡i Zvici 2 Øviv fvM Kiæb|
03. `yBwU msL¨vi †hvMdj 60 Ges we‡qvMdj 10 n‡j, eo msL¨vwU KZ? BangladeshkrshiBank(DataEatryOperator):18
35 40 30 45 DËi: K
†hvMdj I we‡qvMdj †hvM K‡i 2 Øviv fvM Ki‡j eo msL¨vwU cvIqv hv‡e- (60 + 10)  2 = 35|
 civgk©t G RvZxq mgm¨v¸‡jv gy‡L gy‡L mgvavb KivB fv‡jv| †hgb- 60 Gi mv‡_ 10 †hvM Ki‡j nq 70 Ges 70
Gi A‡a©K 35|
04. `ywU msL¨vi †hvMdj 33 Ges we‡qvMdj 15| †QvU msL¨vwU KZ? BangladeshBankAsst.Director:14
9 12 15 18 DËi: K
33 †_‡K 15 we‡qvM Ki‡j 18 Ges 18 Gi A‡a©K 9|
05. `ywU msL¨vi †hvMdj 21, we‡qvMdj 7| eo msL¨vi A‡a©K KZ? RbZve¨vsKwmwbqiAwdmvi:11; PubaliBankLtd.JuniorOfficer
(cash):12
7 6 9 13 DËi: K
21 Gi mv‡_ 7 †hvM Ki‡j nq 28 Ges 28 Gi A‡a©K 14 n‡”Q eo msL¨v| cÖ‡kœ †P‡q‡Q eo msL¨vi A‡a©K,
ZvB 14 Gi A‡a©K n‡e 7|
 †R‡b wbb -05
`ywU msL¨vi ¸Ydj xy †_‡K x I y msL¨v `ywU †ei Kiv wbqgt cÖ_‡g wPšÍv Ki‡eb ¸YdjwU‡Z x I y
KZfv‡e Av‡Q| †hgb- hw` 20 †K a‡i †bqv nq, Zvn‡j 20G x I y Av‡Q- 1  20 = 20, 2  10 = 20,
4  5 = 20 A_©vr, 20 G x I y msL¨vhyMj Av‡Q wZbwU| Gici G‡`i gvS †_‡K cvi‡d± msL¨vhyMjwU Lyu‡R
wb‡Z n‡e| cÖkœ n‡”Q cvi‡d± msL¨vhyMj †KvbwU? cvi‡d± msL¨vhyMj n‡”Q †h msL¨vhyMjwU cÖ‡kœi kZ©‡K wm× K‡i|
 GB AvBwWqvwU GKwU g¨vwRK AvBwWqv, Avcbvi AwfÁZv hZ †ewk n‡e, Avcwb GB AvBwWqvwU e¨envi Ki‡Z ZZ
†ewk gRv cv‡eb|
06. `ywU msL¨vi †hvMdj 17 Ges ¸Ydj 72| †QvU msL¨vwU KZ? ivóªvqZe¨vsKwmwbqiAwdmvi:00
8 9 10 11 DËi: K
g‡bKwi, eo msL¨vwU x Ges †QvU msL¨vwU y
x + y = 17----- (1) xy = 72 ev, x =
y
72
----- (2)
(1) G x Gi gvb ewm‡q cvB,
y
72
+ y = 17 ev, 17
72 2


y
y
ev, y2
 17y + 72 = 0
ev, y2
 9y  8y + 72 = 0 ev, y ( y  9)  8 (y  9) = 0 ev, ( y  9) (y  8) = 0
y = 9 A_ev y = 8
hw` y = 9 nq, Zvn‡j x =
9
72
= 8
hw` y = 8 nq, Zvn‡j x =
8
72
= 9
cÖkœvbymv‡i x n‡”Q eo msL¨v Ges y n‡”Q †QvU msL¨v| ZvB x = 9 Ges y = 8-B †hŠw³K|
AZGe, †QvU msL¨vwU n‡”Q 8|
mgvavb
N
M
L
K
mgvavb
N
M
L
K
mgvavb
N
M
L
K
mgvavb
N
M
L
K
mgvavb
N
M
L
K

Math Tutor 11
 msL¨vhyMj †ei K‡i `ªæZ mgvavb Kiæb- 72 Gi msL¨vhyMj mg~n- 2 I 36, 3 I 24, 4 I 18, 6 I 12, 8 I 9|
GLv‡b cvi‡d± msL¨vhyMj n‡”Q 8 I 9, hv‡ì †hvMdj 17 Ges ¸Ydj 72| myZivs, †QvU msL¨vwU n‡”Q 8|
(Avcbvi g‡b n‡Z cv‡i, meKqwU msL¨vhyMj †ei K‡i mgvavb Ki‡Z †Zv mgq †j‡M hv‡e| GiKg fvevi †Kvb
my‡hvM †bB| KviY GB bvgZv¸‡jv Avcbvi gyL¯’ Av‡Q, ZvB cÖ‡kœi kZ© †`L‡jB e‡j w`‡Z cvi‡eb †Kvb
msL¨vhyMjwU Avcbv‡K P‡qR Ki‡Z n‡e|)
07. `ywU msL¨vi ¸Ydj 189 Ges msL¨v `ywUi †hvMdj 30| msL¨v `ywU KZ? gnvwnmvewbixÿKIwbqš¿‡KiKvh©v.AaxbRywbqiAwWUi:14
9 I 21 7 I 23 8 I 22 22 I 18 DËi: K
Ackb Gi msL¨vhyMjwU cÖ‡kœi kZ©‡K wm× K‡i, ZvB GwUB DËi n‡e|
08. †Kvb `ywU msL¨vi †hvMdj 10 Ges ¸Ydj 24? mnKvix_vbvcwievicwiKíbvAwdmvi:98
4,  6  6,  4 12,  2 4, 6 DËi: N
Ackb Gi msL¨vhyMjwU cÖ‡kœi kZ©‡K wm× K‡i, ZvB GwUB DËi n‡e|
09. `yBwU msL¨vi AšÍi 7 Ges Zv‡ì MyYdj 60| msL¨vØ‡qi GKwU- DBBLAssistantofficer:09/BKBofficer:07
4 5 6 7 DËi: L
60 Gi 5 I 12 msL¨vhyMjwU cÖ‡kœi kZ©‡K wm× K‡i|
PP©v Kiæb
10. `ywU msL¨vi †hvMdj 23 Ges we‡qvMdj 21| †QvU msL¨vwU KZ? Sonali,JanataandAgraniBankLtd.SeniorOfficer:08
4 3 2 None DËi: N
11. `yBwU msL¨vi ¸Ydj 10 Ges Zv‡ì mgwó 7 n‡j, e„nËg msL¨vwU KZ? EXIMBankLtd.Officer:14
 2 2 4 5 DËi: N
12. `ywU msL¨vi ¸Ydj 120 Ges Zv‡ì e‡M©i †hvMdj 289| msL¨vØ‡qi mgwó KZ? EXIMBankLtd.Officer(IT):13
20 21 22 23 DËi: N
13. `yBwU msL¨vi ¸Ydj 42 Ges we‡qvMdj 1 n‡j msL¨v `y&ÕwU KZ?mve-†iwR÷ªvit03
4, 3 7, 6 8, 6 10, 8 DËi: L
14. 2wU msL¨vi †hvMdj 48 Ges Zv‡ì ¸Ydj 432| Z‡e eo msL¨vwU KZ? cwievicwiKíbvAwa`ßiwb‡qvMcixÿv:14
36 37 38 40 DËi: K
01.09 `ywU msL¨vi †hvMdj, we‡qvMdj I ¸Ydj
01. `ywU msL¨vi †hvMdj Zv‡ì we‡qvMd‡ji wZb¸Y| †QvU msL¨vwU 20 n‡j, eo msL¨vwU KZ? evsjv‡`kK…wle¨vsKAwdmvi:11
5 40 60 80 DËi: L
g‡bKwi, eo msL¨vwU = x Ges †QvU msL¨vwU = y
cÖkœg‡Z, x + y = 3(x  y) (we‡qvMdj‡K 3 ¸Y Ki‡j †hvMd‡ji mgvb n‡e)
ev, x + 20 = 3x  320(†QvU msL¨v, y = 20 ewm‡q)
ev, x + 20 = 3x  60 ev, 2x = 80 x = 40|
 Option Test: Ackb mwVK n‡e bv, KviY eo msL¨vwU 20 Gi †P‡q eo n‡e| 40 + 20 = 60 Ges
(40  20)3 = 203 = 60 (k‡Z©i mv‡_ wg‡j †M‡Q) I k‡Z©i mv‡_ wgj‡e bv|
02. `ywU msL¨vi AšÍi 2 Ges mgwó 4| Zv‡ì e‡M©i AšÍi KZ? BangladeshBankAsst.Direefor:12
7 8 6 5 DËi: L
x = (2 + 4)  2 = 3 Ges y = (4  2)  2 = 1 x2
 y2
= 32
 12
= 9  1 = 8|
A_ev x2
 y2
= (x + y) (x  y) = 42 =8 (exRMwY‡Z wbq‡g GB mgm¨vwU mgvavb Kiv AwaKZi mnR)
03. `yBwU msL¨vi mgwó 40 Ges Zv‡ì AšÍi 4| msL¨vØ‡qi AbycvZ KZ? JanataBankLld.ExecutiveOffice(Morring):17
11 : 9 11 : 18 21 : 19 22 : 9 DËi: K
x = 40 + 4 = 44  2 = 22, y = 40  4 = 36  2 = 18 (GB MYbv¸‡jv gy‡L gy‡L K‡i †dj‡eb)
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12Math Tutor
myZivs, msL¨vØ‡qi AbycvZ = 22 : 18 = 11 : 9 |
 †R‡b wbb-06 (`ye©j‡ì Rb¨)
 (x + y) 
2
1
= 51 ev, x + y = 51  2 = 102 A_ev (x  y) 
2
1
= 5 ev, x  y = 5  2 = 10
 Tips: A‡a©K _vK‡j wØ¸Y Ki‡jB x+y/ x -y Gi gvb cvIqv hvq| GKBfv‡e GK-Z…Zxqvsk _vK‡j wZb¸Y, GK PZz_©vsk
_vK‡j 4¸Y, GK cÂgvsk _vK‡j 5 ¸Y Ki‡j x + y/ x - y Gi gvb cvIqv hvq|
04. `ywU msL¨vi A‡a©‡Ki †hvMdj 51| Zv‡ì cv_©‡Kï GK PZz_©vsk 13| msL¨vØq KZ? Dc-mnKvixcwiPvjK(kÖg):01
52, 70 26, 27 25, 66 77, 25 DËi: N
(we¯ÍvwiZ)
 ey‡S ey‡S mgvavb: `ywU msL¨vi A‡a©‡Ki †hvMdj 51, Gevi A‡a©K‡K wØ¸Y Ki‡j msL¨v `ywUi †hvMdj cvIqv hv‡e|
A_©vr, x + y = 51  2 = 102| msL¨v `ywUi cv_©‡Kï GK PZz_©vsk 13, Gevi GK PZz_©vsk‡K 4 ¸Y Ki‡j msL¨v
`ywUi we‡qvMdj cvIqv hv‡e| A_©vr, x - y = 13 4 = 52|
 x = 102 + 52 = 154 Gi A‡a©K 77 Ges y = 102 - 52 = 50 Gi A‡a©K 25|
05. `ywU msL¨vi cv_©K¨ 11| Zv‡ì †hvMd‡ji GK cÂgvsk 9| msL¨v `ywU wK wK? evsjv‡`k e¨vsK (GwW) 2014
28 Ges 17 29 Ges 18 30 Ges 19 †Kv‡bvwUB bq DËi: K
x - y = 11Ges x + y = 9  5 = 45(GK cÂgvsk 9 †K 5 ¸Y Kiv n‡q‡Q)
 x = 45 + 11 = 56 Gi A‡a©K 28 Ges y =45 - 11 = 34 Gi A‡a©K 17|
 †R‡b wbb-07 (wb‡Pi mgm¨v¸‡jv exRMvwYwZK m~Î cÖ‡qvM K‡iI mn‡RB mgvavb Ki‡Z cv‡ib)
 (x+y)2
= x2
+ 2xy + y2
 (x-y)2
= x2
- 2xy + y2
x2
- y2
= (x + y) (x-y)
06. `ywU msL¨vi e‡M©i mgwó 80 Ges Zv‡ì cv_©‡Kï eM© 16| msL¨vØ‡qi ¸Ydj KZ?UCBL wmwbqi Awdmvi 2011
10 16 30 32 DËi: N
†`qv Av‡Q, x2
+ y2
= 80Ges (x-y)2
= 16
Avgiv Rvwb,(x-y)2
= x2
+ y2
- 2xy ev, 16 = 80 - 2xy ev, 2xy = 64  xy = 32|
cÖ‡kœ hw` msL¨v `ywU Rvb‡Z PvIqv nZ? Zvn‡j 32 †_‡K x I y msL¨vhyMj‡K †ei K‡i wb‡Z n‡e| 32G x I y
Gi Rb¨ wZbwU msL¨vhyMj Av‡Q| †hgb- 1 I 32, 2 I 16 , 4 I 8 (GLv‡b cÖ‡Z¨KwU msL¨vhyM‡ji ¸Ydj 32)
GB wZbwU msL¨v hyM‡ji gv‡S ïay 4 I 8 hyMjwU cÖ‡kœi kZ© c~Y© K‡i| A_©vr, 42
+ 82
= 80 Ges 8 - 4 = 4 Gi
eM© 16|
 `ywU msL¨vi ¸Ydj †_‡K cvi‡d± msL¨vhyMj †ei Kivi †KŠkjwU fv‡jvfv‡e Avq‡Ë¡ Ki‡Z cvi‡j Avcwb A‡bK
RvqMvq `viæY myweav cv‡eb| cvi‡d± msL¨vhyMj n‡”Q H msL¨vhyMj †hwU cÖ‡kœi kZ©‡K c~Y© K‡i|
07. `ywU msL¨vi mgwó 15 Ges Zv‡ì e‡M©i mgwó 113| msL¨v `ywU †ei Kiæb| RbZv e¨vsK wj. (AEO) 2015
6 Ges 9 7 I 8 10 I 5 †Kv‡bvwUB bq DËi: L
†`qv Av‡Q, x + y = 15Ges x2
+ y2
= 113|
Avgiv Rvwb, (x+y)2
= x2
+ 2xy + y2
ev, 152
= 113 + 2xy ev, 225 - 113 = 2xy
xy =
2
112
= 56(Dc‡ii A‡¼i gZ hw` msL¨v `ywUi ¸Ydj Rvb‡Z PvBZ, Zvn‡j 56 B DËi nZ, wKš‘ msL¨v `ywU
†ei Ki‡Z e‡j‡Q ZvB cvi‡d± msL¨vhyMj †ei Ki‡Z n‡e| )
56 G x I y Gi gvb wn‡m‡e wZbwU msL¨vhyMj 2 I 28, 4 I 14, 7 I 8 Av‡Q| G‡ì gv‡S ïay 7 I 8
msL¨vhyMjwU cÖ‡kœi kZ©‡K wm× K‡i| myZivs msL¨v `ywU n‡”Q 7 I 8|
 Avcwb GKevi welqwU eyS‡Z cvi‡j †Kvb msL¨v †`Lv gvÎB Zvi gv‡S cvi‡d± msL¨vhyMj †`L‡Z cv‡eb| mewKQz
AwfÁZvi Dci wbf©i K‡i|
08. `yBwU msL¨vi AšÍi 5 Ges Zv‡ì e‡M©i cv_©K¨ 65| eo msL¨vwU KZ? evsjv‡`k nvDR wewìs dvBb¨vÝ K‡cv©‡ikb 2017
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Math Tutor 13
13 11 8 9 DËi: N
x - y = 5 Ges x2
- y2
= 65
Avgiv Rvwb, x2
- y2
= (x+y) (x-y) = 65 ev, (x + y)5 = 65 ev, x + y =
5
65
= 13(cÖ‡kœ x - y = 5
†Ìqv Av‡Q Ges Avgiv cvBjvg x + y = 13 | Gevi †hvMdj I we‡qvMdj †hvM K‡i 2 Øviv fvM Ki‡j eo msL¨vwU cvIqv
hv‡e)
 eo msL¨vwU =
2
5
13 
= 9|
01.10 M.mv.¸ †_‡K msL¨v wbY©q
 †R‡b wbb - 07
†h msL¨vwU `ywU ¸Yd‡ji `ywU‡ZB _v‡K †m msL¨vwUB M.mv.¸| `ywU msL¨vi ¸Ydj †_‡K msL¨vwU †ei Kivi `ÿZv hZ
†ewk n‡e G RvZxq mgm¨v mgvavb Kiv ZZ mnR n‡e|
01. cÖ_g I wØZxq msL¨vi ¸Ydj 42 Ges wØZxq I Z…Zxq msL¨vi ¸Ydj 49| wØZxq msL¨vwU KZ? cÖv_wgKmnKvixwkÿK
(gyw³‡hv×v)knx`gyw³‡hv×vimšÍvb):10(†ngšÍ)
5 6 7 8 DËi: M
cÖ_g  wØZxq msL¨v =42 Ges wØZxq I Z…Zxq msL¨vi ¸Ydj = 49| G‡ì M.mv.¸ 7-B n‡e wØZxq msL¨vwU,
KviY `ywU ¸Yd‡jB wØZxq msL¨vwU common Av‡Q|  wØZxq msL¨vwU 7|
02. cÖ_g I wØZxq msL¨vi ¸Ydj 35 Ges wØZxq I Z…Zxq msL¨vi ¸Ydj 63| wØZxq msL¨vwU KZ? cwiKíbvgš¿YvjqWvUv
cÖ‡mwms Acv‡iUi:02
5 6 7 8 DËi: M
35 = 5  7 Ges 63 = 7  9|  wØZxq msL¨vwU 7|
03. wZbwU cici †gŠwjK msL¨vi cÖ_g `yBwU msL¨vi ¸Ydj 91, †kl `yBwUi ¸Ydj 143 n‡j, msL¨v wZbwU KZ? moKI
Rbc_Awa`ß‡iiDcmnKvixcÖ‡KŠkjx:10
7, 13, 11 7, 11, 13 11, 7, 13 11, 13, 7 DËi: K
91 = 7  13 Ges 143 = 11  13 G‡ì M.mv.¸ = 13|  msL¨v 3wU n‡”Q 7, 11 Ges 13|
01.11 ¯^Kxq gvb I ¯’vbxq gvb
(i) ¯^Kxq gvb (Face Value) t †Kvb mv_©K A¼ Avjv`vfv‡e wjL‡j †h msL¨v cÖKvk K‡i, Zv A‡¼i ¯^Kxq gvb|
(ii) ¯’vbxq gvb (Place/local Value) t K‡qKwU A¼ cvkvcvwk wjL‡j †Kvb mv_©K A¼ Zvi Ae¯’v‡bi Rb¨ †h
msL¨v cÖKvk K‡i, Zv‡K H A‡¼i ¯’vbxq gvb e‡j|
Place Value Chart (¯’vbxq gvb wbY©‡qi QK)
†KvwU wbhyZ jÿ AhyZ nvRvi kZK `kK GKK ¯’vbxq gvb
9 2 8 3 2 5 4 7 71 = 7
410= 40
5100 = 500
21000 =2000
310000 = 30000
810000 = 800000
21000000 = 2000000
910000000 = 90000000
01. 666 msL¨vwU‡Z me©ev‡gi 6 Gi gvb KZ? cwievicwiKíbvwnmvei¶K/¸`vgi¶K/†Kvlva¨¶:11
60 600 6 DËi: L
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14Math Tutor
me©ev‡gi 6 ÔkZKÕ ¯’v‡bi A¼ nIqvq Gi gvb n‡e 600|
02. 3254710 msL¨vwU‡Z 5 Gi ¯’vbxq gvb KZ? QuantitativeAptitudebyS.Chand&Aggarwal
5 10000 50000 54710 DËi: M
03. 458926 msL¨vwU‡Z 8 Gi ¯^Kxq gvb KZ? QuantitativeAptitudebyS.Chand&Aggarwal;Pubali
Bank,JuniorOfficer-2019
8 1000 8000 8926 DËi: K
04.503535 msL¨vwU‡Z 3 Gi ¯’vbxq gvb mg~‡ni mgwó KZ? QuantitativeAptitudebyS.Chand&Aggarwal
6 60 3030 3300 DËi: M
mn¯ª ¯’v‡bi 3 Gi gvb 3000 I `k‡Ki ¯’v‡bi 3 Gi gvb 30|  mgwó = 3000 + 30 = 3030|
05. 527435 msL¨vwU‡Z 7 I 3 Gi ¯’vbxq gv‡bi g‡a¨ cv_©K¨ KZ? QuantitativeAptitudebyS.Chand&Aggarwal
4 5 45 6970 DËi: N
7 I 3 Gi ¯’vbxq gv‡bi cv_©K¨ = 7000 - 30 = 6970|
06. 32675149 msL¨vwU‡Z 7 Gi ¯’vbxq gvb I ¯^Kxq gv‡bi g‡a¨ cv_©K¨ KZ? QuantitativeAptitudebyS.Chand&
Aggarwal; Pubali Bank Ltd. Senior Offficer/Officer :16
5149 64851 69993 75142 DËi: M
cÖ`Ë msL¨vwU 32675149 †_‡K 7 Gi ¯^Kxq gvb (face value)I ¯’vbxq gvb (local value) ‡ei
Ki‡Z n‡e| Zvici G‡`i †h cv_©K¨ (difference) †ei n‡e ZvB DËi|
32675149msL¨vwU‡Z 7 Gi ¯^Kxq gvb 7 Ges ¯’vbxq gvb 70000. Zv‡`i cv_©K¨ = (70000 - 7) = 69993.
07. cvuP A‡¼i e„nËg I ÿz`ªZg msL¨vi mgwó KZ? QuantitativeAptitudebyS.Chand &Aggarwal
1,110 10,999 109,999 111,110
08. cvuP A¼wewkó ÿz`ªZg msL¨v †_‡K wZb A¼wewkó e„nËg msL¨v we‡qvM Ki‡j KZ Aewkó _v‡K? Quantitative
AptitudebyS.Chand&Aggarwal
1 9000 9001 90001 DËi: M
09. 3 w`‡q ïiæ I 5 w`‡q †kl nIqv 5 A¼wewkó ÿz`ªZg msL¨vwU KZ n‡e? QuantitativeAptitudebyS.Chand&
Aggarwal
31005 30015 30005 30025 DËi: M
10. 2, 4, 0, 7 A¼¸‡jv Øviv MwVZ 4 A‡¼i ÿz`ªZg msL¨v †KvbwU? QuantitativeAptitudebyS.Chand&Aggarwal
2047 2247 2407 2470 DËi: K
11. GKwU msL¨vi kZK, `kK I GKK ¯’vbxq AsK h_vµ‡g p, q, r n‡j msL¨vwU n‡e-†Rjvwbev©PbAwdmvit04
100r + 10p + q 100p + 10q + r
100q + 10r + p 100pq + r DËi: L
p q r
r 1 = r
q 10 = 10q
p 100 = 100p
msL¨vwU = 100p + 10q + r
12. 856973 msL¨vwU‡Z 6 Gi ¯’vbxq gvb I ¯^Kxq gv‡bi g‡a¨ cv_©K¨ KZ? Pubali Bank Ltd. Trainee Asst.
Teller : 17; Probashi Kallyan Bank Ltd. Senior Officer : 14
973 6973 5994 None of these DËi : M
13. 2, 3 Ges 4 Øviv 3 A‡¼i KZwU we‡Rvo msL¨v MVb Kiv hvq?gv`K`ªe¨ wbqš¿Y Awa`ß‡ii mnKvix cwiPvjK-2013
2wU 5wU 6wU 7wU
2, 3, 4 Øviv 3 A‡¼i 2wU we‡Rvo msL¨v MVb Kiv hvq| †hgb- 243 Ges 423| DËi: K
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Math Tutor 15
01.12 A¼Ø‡qi ¯’vb cwieZ©b msµvšÍ mgm¨v
 †R‡b wbb – 08 (AwZ `ye©j‡`i Rb¨)
 `yB A¼ wewkó msL¨v (Original number) I ¯’vb wewbgq msL¨v (Reversed number) :
xy Gi ¯’vbxq gvb wbY©q Kiv hvK| GLv‡b, x n‡”Q GKK ¯’vbxq A¼ Ges y n‡”Q `kK ¯’vbxq A¼|
∴yx
x1 = x
y 10 = 10y
∴ yx Gi ¯’vbxq gvb = 1oy + x
GLb hw` A¼ `ywU ¯’vb wewbgq K‡i Zvn‡j †h bZzb msL¨vwU MwVZ n‡e Zvi ¯’vbxq gvb cwieZ©b n‡e|
ZLb bZzb msL¨vwU‡Z y P‡j hv‡e GKK ¯’v‡b Ges x P‡j hv‡e `kK ¯’v‡b| A_©vr,
∴xy
y1 = y
x 10 = 10x
∴ yx Gi A¼Øq ¯’vb wewbgq Kivi ci xy Gi ¯’vbxq gvb = 10x + y
 hw` cÖkœc‡Î Ô`yB A¼ wewkó msL¨vi A¼Ø‡qi mgwóÕ †`qv _v‡K Zvn‡j `yB A¼ wewkó msL¨v (Original
number) I ¯’vb wewbgqK…Z msL¨v (Reversed number) wb‡Pi wbq‡g a‡i wb‡eb, Zvn‡j mgm¨vwU mn‡R
mgvavb Kiv hv‡e| Ô `yB A¼ wewkó msL¨vi A¼Ø‡qi mgwó 7Õ GB D`vniYwU mvg‡b †i‡L welqwU eySv‡bv hvK|
g‡bKwi, GKK ¯’vbxq A¼ x n‡j `kK ¯’vbxq A¼ (7  x)
msL¨vwU = 10 (7  x) + x = 70 - 9x
¯’vb wewbgq Ki‡j msL¨vwU `vovq = 10x + (7  x) = 9x + 7|
01. `yB A¼ wewkó msL¨vi A¼Ø‡qi mgwó 10| msL¨vwU †_‡K 18 we‡qvM Ki‡j A¼Øq ¯’vb cwieZ©b K‡i| msL¨vwU
KZ? ÷¨vÛvU© e¨vsK wj. (cÖ‡ekbvwi Awdmvi) 2008
64 46 55 73 DËi: K
g‡bKwi, GKK ¯’vbxq A¼ x n‡j `kK ¯’vbxq A¼ (10  x)
msL¨vwU = 10 (10  x) + x = 100 - 9x
¯’vb wewbgq Ki‡j msL¨vwU `vovq = 10x + (10  x) = 9x + 10
kZ©g‡Z, (100 - 9x) – 18 = 9x + 10 ev, 9x + 9x = 100 – 28 ev, 18x = 72  x = 4
myZivs, wb‡Y©q msL¨vwU = 100 – 9x = 100 – 94 = 100 – 36 = 64 |
 Option Test : 64 – 18 = 46
02. `yB A¼ wewkó msL¨vi A¼Ø‡qi mgwó 10| msL¨vwU †_‡K 72 we‡qvM Ki‡j A¼Øq ¯’vb cwieZ©b K‡i| msL¨vwU
KZ? AvBwmwe A¨vwm‡÷›U †cÖvMÖvgvi 2008
82 91 55 37 DËi: L
Option Test : 91 – 72 = 19
03. `yB A¼ wewkó GKwU msL¨vi A¼Ø‡qi ¸Ydj 8| msL¨vwUi mv‡_ 18 †hvM Ki‡j A¼Øq ¯’vb cwieZ©b K‡i|
msL¨vwU KZ? hgybv e¨vsK wj. (GgwUI) 2012
18 24 42 81 DËi: L
g‡bKwi, GKK ¯’vbxq A¼ x Ges `kK ¯’vbxq A¼ y  msL¨vwU = 10y + x
A¼Øq ¯’vb wewbgq Ki‡j msL¨vwU `uvovq = 10x + y
1g kZ©g‡Z, xy = 8 ---- (i)
2q kZ©g‡Z, (10y + x) + 18 = 10x + y
ev, 9x – 9y = 18 ev, 9(x – y) = 18 ev, x – y = 2  x = 2 + y ---- (ii)
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16Math Tutor
(i) bs mgxKi‡Y x Gi gvb ewm‡q cvB, (2 + y) y = 8 ev, 2y + y2
= 8 ev, y2
+ 2y – 8 = 0
ev, y2
+ 4y – 2y – 8 = 0 ev, y (y + 4) – 2 ( y + 4) = 0 ev, (y + 4) (y – 2) = 0
y + 4 = 0 A_ev y – 2 = 0
 y = – 4 (y Gi FYvZ¥K gvb MÖnY‡hvM¨ bq) y = 2
(ii) bs mgxKi‡Y y =2 ewm‡q cvB, x = 2 + 2 = 4
myZivs msL¨vwU = 10y + x = 10 2 + 4 = 24
 Option Test: me KqwU Ack‡bi A¼Ø‡qi ¸Ydj 8, ZvB 1g kZ©wU cÖgvY Kivi `iKvi †bB|
Ackb  24 + 18 = 42
 †R‡b wbb – 09
 g¨vwRK Z_¨- (Aek¨B Av‡jvPbvwU fv‡jvfv‡e eyS‡eb, cÖ‡qvR‡b GKvwaKevi coyb|)
(1) Original number 37  Original number I Reversed number Gi
cv_©K¨ memgq 9 Øviv wefvR¨ n‡e|
 Original number I Reversed number Gi
cv_©K¨‡K 9 Øviv fvM K‡i cÖvß fvMdj = msL¨vwUi
A¼Ø‡qi AšÍi/cv_©K¨| so, cÖkœc‡Î Original I
Reverse number Gi cv_©K¨ †`qv _vK‡j Zv‡K 9 Øviv
fvM Ki‡j A¼Ø‡qi AšÍi cvIqv hv‡e|
(2) Reversed number 73
(3) Difference 73  37 = 36
(4) Divide by 9
9
36
= 4
(5) Difference of 2 digits 7  3 = 4
 Original number Gi GKK ¯’vbxq A¼ eo n‡j reverse Kivi ci gvb e„w× cv‡e| †hgb-
34 (original number)  43(GLv‡b original number Gi GKK ¯’vbxq A¼ 4 eo nIqvq gvb e„w× n‡q‡Q| c~‡e©i
34 †_‡K 9 e„w× †c‡q 43 n‡q‡Q| GRb¨ e„w× cvIqvi K_v ej‡j original number Gi GKK ¯’vbxq A¼ eo _vK‡e|)
Ges Original number Gi GKK ¯’vbxq A¼ †QvU n‡j, reverse Kivi ci gvb n«vm cvq| †hgb-
43(original number) 34 (GLv‡b original number Gi GKK ¯’vbxq A¼ 3 †QvU nIqvq gvb n«vm †c‡q‡Q|
c~‡e©i 43 †_‡K 9n«vm †c‡q 34 n‡q‡Q| GRb¨ e„w× cvIqvi K_v ej‡j original number Gi GKK ¯’vbxq A¼ †QvU
_vK‡e|)
 (A_ev) reverse Kivi ci gvb e„w× †c‡j reversed number wU eo n‡e wKš‘ original number wU †QvU
n‡e Ges GKBfv‡e reverse Kivi ci gvb n«vm †c‡j reversed number wU †QvU n‡e wKš‘ original number
wU eo n‡e| (†k‡li wbqgwUB †ek mnR)
04. `yB A¼ wewkó GKwU msL¨vi A¼Ø‡qi mgwó 7| A¼Øq ¯’vb wewbgq Ki‡j †h msL¨v cvIqv hvq, Zv cÖ`Ë msL¨v
†_‡K 9 †ewk| msL¨vwU KZ? e¨vsKviÕm wm‡jKkb KwgwU (wmwbqi Awdmvi) 2018
61 25 34 43 DËi: M
g‡bKwi,GKK ¯’vbxq A¼ x, Zvn‡j `kK ¯’vbxq A¼ n‡e 7  x |
 msL¨vwU = 10 (7  x) + x = 70  9x
A¼ `ywU ¯’vb wewbgq Ki‡j msL¨vwU = 10x + (7  x) = 9x + 7
cÖkœg‡Z, 9x + 7  9 = 70 - 9x (Reversed msL¨vwU original msL¨v †_‡K 9 †ewk nIqvq 9 we‡qvM K‡i mgvb Kiv n‡q‡Q)
ev, 9x + 9x = 70 + 2 ev 18x = 72x = 4
myZivs msL¨vwU = 70  9x = 70  9  4 = 70  36 = 34|
 Original number I Reversed number Gi cv_©K¨ 9 ‡K 9 Øviv fvM K‡i 1 cvIqv hv‡”Q, GB 1 n‡”Q original
msL¨vwUi A¼Ø‡qi cv_©K¨| Zvn‡j cÖ`Ë Ackb¸‡jvi gv‡S †hwUi A¼Ø‡qi cv_©K¨ 1 Av‡Q †mwUB n‡e wb‡Y©q msL¨vwU| Avgiv
Ackb I †Z `ywU‡Z A¼Ø‡qi cv_©K¨ 1 †`L‡Z cvw”Q| Avgiv Rvwb Original number I Reversed
number Gi cv_©K¨ †ewk/e„w× †c‡j Original number Gi GK ¯’vbxq A¼ eo nq| †h‡nZz GB cÖ‡kœ †ewk/
e„w×i K_v ejv n‡q‡Q, ZvB Original number wUi GK ¯’vbxq A¼ eo n‡e| G Abymv‡i Ackb I Gi
gv‡S mwVK DËi n‡e | (GB wbqgwU eyS‡Z mgq jvM‡jI Gi gva¨‡g me‡P‡q Kg mg‡q mgvavb Kiv hvq)
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Math Tutor 17
 (A_ev) 6116 (cv_©K¨ 45, hv mwVK bq) 25 52 (cv_©K¨ 27, hv mwVK bq) 34 43
(cv_©K¨ 9, cÖ‡kœ †h‡nZ z reverse Kivi ci reversed number wU eo n‡”Q †m‡nZz original number wU
†QvU n‡e| Ackb -†Z orginal number wU †QvU weavq GwUB mwVK DËi)
43 34 (cv_©K¨ 9 _vK‡jI original number wU eo nIqvq GwU mwVK DËi bq) |
05. `yB AsK wewkó GKwU msL¨v, AsKØ‡qi ¯’vb wewbg‡qi d‡j 54 e„w× cvq| AsK `ywUi †hvMdj 12 n‡j msL¨vwU
KZ? 37Zg wewmGm
57 75 39 93 DËi: M
54 ÷ 9 = 6, wb‡Y©q msL¨vwUi A¼Ø‡qi cv_©K¨ n‡e 6, hv Ackb I †Z Av‡Q| wKš‘ cÖ‡kœ Ôe„w×Õ
K_vwU ejv _vKvq msL¨vwUi GKK ¯’vbxq A¼wU ÔeoÕ n‡e †m Abymv‡i mwVK DËi |
 (A_ev) 39  93 (cv_©K¨ 54, cÖ‡kœ Ôe„w×Õ ejvq reversed number wU eo n‡e Ges original number
wU †QvU n‡e, ZvB GwUB mwVK DËi)|
06. `yB A¼ wewkó GKwU msL¨vi A¼Ø‡qi ¯’vb cwieZ©b K‡i cÖvß msL¨v g~j msL¨v A‡cÿv 54 Kg| msL¨vwUi A¼Ø‡qi
mgwó 12 n‡j, g~j msL¨vwU KZ?
28 39 82 †KvbwUB bq DËi: N
g‡bKwi, GKK ¯’vbxq A¼ x, Zvn‡j `kK ¯’vbxq A¼ n‡e 12  x |
 msL¨vwU = 10 (12  x) + x = 120  9x
A¼ `ywU ¯’vb wewbgq Ki‡j msL¨vwU = 10x + (12  x) = 9x + 12
cÖkœg‡Z, 9x +12  54 = 120  9x (Reversed msL¨vwU originalmsL¨v †_‡K 54Kg nIqvq 54 †hvM K‡i mgvb Kiv n‡q‡Q)
ev, 9x + 9x = 120  66 ev, 18x = 54x = 3
myZivs msL¨vwU = 120  93 = 120  27 = 93
 cÖkœvbymv‡i A¼Ø‡qi mgwó n‡Z n‡e 12 hv ïay Ackb †Z Av‡Q Ges 54÷9 = 6 Abymv‡i A¼Ø‡qi cv_©K¨
n‡Z n‡e 6| wKš‘ cÖ‡kœ Reversed number g~j msL¨v (Original number) A‡cÿv ÔKgÕ nIqvq msL¨vwUi GKK
¯’vbxq A¼ Ô‡QvUÕ n‡Z n‡e, hv Ack‡b †bB ZvB GB AckbwUI mwVK bq| Z‡e GKK ¯’vbxq A¼ Ô‡QvUÕ (A_©vr,
93 n‡j)n‡j DËiwU mwVK nZ|
 Ackb , I Gi Original number I Reversed number Gi cv_©K¨ 54, G‡`i gv‡S Ackb I Gi
GKK ¯’vbxq A¼ eo nIqvq Giv ev` hv‡e| Ackb Gi GKK ¯’vbxq A¼ Ô‡QvUÕ n‡jI cÖkœvbymv‡i Gi A¼Ø‡qi
mgwó 12 bq, ZvB GwUI evwZj| DËi n‡e Ô‡KvbwUB bqÕ|
07. `yB A¼ wewkó †Kvb msL¨vi A¼Ø‡qi mgwó 9| A¼ `ywU ¯’vb wewbgq Ki‡j †h msL¨v cvIqv hvq Zv cÖ`Ë msL¨v n‡Z
45 Kg| msL¨vwU wbY©q Kiæb| gva¨wgK mnKvix cÖavb wkÿK I †Rjv mnKvix wkÿv Awdmvi 2003
54 63 72 81 DËi: M
45 ÷ 9 = 5, wb‡Y©q msL¨vwUi A¼Ø‡qi cv_©K¨ n‡e 5, hv ïay Ackb †Z Av‡Q|
 (A_ev) 72  27 (cv_©K¨ 45, Ab¨‡Kvb Ack‡bi cv_©K¨ 45 bv _vKvq mivmwi GwUB DËi n‡e)
hw` cÖ‡kœ AviI GKwU Ackb 27 _vKZ, Zvn‡jI DËi 72-B nZ| KviY cÖ‡kœ reversed number wU original
number †_‡K ÔKg/‡QvUÕ nIqvq original number wU eo n‡e|)
08. `yB A¼ wewkó GKwU msL¨vi A¼Ø‡qi mgwó 8| A¼Øq ¯’vb wewbgq Ki‡j †h msL¨v cvIqv hvq Zv cÖ`Ë msL¨v n‡Z
54 Kg| msL¨vwU KZ? RbZv e¨vsK (Awdmvi) 2009
71 80 62 53 DËi: K
54÷ 9 = 6, wb‡Y©q msL¨vwUi A¼Ø‡qi cv_©K¨ n‡e 6, hv ïay Ackb †Z Av‡Q|
 (A_ev) 71  17 (cv_©K¨ 54, hv Ab¨‡Kvb Ack‡b †bB, ZvB mivmwi GwUB DËi)
09. `yB A¼ wewkó GKwU msL¨vi A¼Ø‡qi mgwó 9| msL¨vwU n‡Z 9 we‡qvM Ki‡j Gi A¼Øq ¯’vb wewbgq K‡i| msL¨vwU
KZ? cwievi cwiKíbv Awa`ßi cwi`wk©Kv cÖwkÿYv_x© 2013
34 67 54 23 DËi: M
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18Math Tutor
9 ÷ 9 = 1, wb‡Y©q msL¨vwUi A¼Ø‡qi cv_©K¨ n‡e 1, hv me KqwU Ack‡bB Av‡Q| wKš‘ original number
†_‡K 9 we‡qvM K‡i reversed number Avm‡e ZvB reversed number wU †QvU n‡e Ges original number
wU eo n‡e| G Abymv‡i ïay Ackb †K reverse Ki‡j reversed number wU †QvU n‡e|
10. `yB A¼ wewkó †Kv‡bv msL¨v Ges A¼Øq ¯’vb wewbgq Ki‡j MwVZ msL¨vi cv_©K¨ 36| msL¨vwUi A¼Ø‡qi AšÍi
KZ? evsjv‡`k e¨vsK (A¨vwm÷¨v›U wW‡i±i) 2012
4 2 10 16 DËi: K
GKK ¯’vbxq A¼ x Ges `kK ¯’vbxq A¼ y n‡j msL¨vwU = 1oy + x, A¼Øq ¯’vb wewbgq Ki‡j msL¨vwU `uvovq
10x + y . (Original number I reversed number Gi cv_©K¨ n‡”Q 36)
kZ©g‡Z, (10x + y) – (10y + x) = 36
ev, 9x – 9y = 36
ev, 9 (x – y) = 36
 x – y =
36
9
= 4 myZivs msL¨vwUi A¼Ø‡qi AšÍi 4 |
 kU©Kv‡U mgvavbt msL¨vwUi A¼Ø‡qi AšÍi =
36
9
= 4
11. `yB A¼ wewkó GKwU abvZ¥K c~Y©msL¨v Ges A¼Øq ¯’vb wewbgq Ki‡j MwVZ msL¨vi cv_©K¨ 27| msL¨vwUi A¼Ø‡qi
AšÍi KZ? evsjv‡`k K…wl e¨vsK wj. (wmwbqi Awdmvi) 2017
3 4 5 6 DËi: K
msL¨vwUi A¼Ø‡qi AšÍi =
27
9
= 3
12. `yB A¼wewkó GKwU msL¨v msL¨vwUi A¼Ø‡qi †hvMd‡ji 4 ¸Y| msL¨vwUi mv‡_ 27 †hvM Ki‡j A¼Øq ¯’vb wewbgq
K‡i| msL¨vwU KZ? evsjv‡`k wkwcs K‡cv©‡ikb 2018
12 42 24 36 DËi: N
36 = 3 + 6 = 9  4= 36,  36 + 27 = 63
13. `yB A¼ wewkó GKwU msL¨vi `kK ¯’vbxq A¼wU GKK ¯’vbxq A¼ †_‡K 5 eo| msL¨vwU †_‡K A¼Ø‡qi mgwói 5
¸Y we‡qvM Ki‡j A¼Øq ¯’vb wewbgq K‡i| msL¨vwU KZ? _vbv mnKvix wkÿv Awdmvi : 2005
61 94 72 83 DËi: M
cÖ`Ë Ackbmg~‡ni cÖ‡Z¨KwU‡Z `kK ¯’vbxq A¼wU GKK ¯’vbxq A¼ †_‡K 5 eo, ZvB GB kZ©wU cÖgvY Kivi
`iKvi †bB| Avgiv 2q kZ©wU cÖgvY Kie-
61  A¼Ø‡qi mgwó = 6 + 1 = 7, mgwói 5 ¸Y = 75 = 35|  61 - 35 = 26 (GwU mwVK bq)
94  A¼Ø‡qi mgwó = 9 + 4 = 13, mgwói 5 ¸Y = 135 = 65|  94 - 65 = 29 (GwU mwVK bq)
72  A¼Ø‡qi mgwó = 7 + 2 = 9, mgwói 5 ¸Y = 95 = 45 |  72 - 45 = 27 (GwU mwVK)
14. `yB A¼ wewkó GKwU msL¨vi GK‡Ki A¼ `k‡Ki A¼ A‡cÿv 3 †ewk| msL¨vwU Gi A¼Ø‡qi mgwói wZb¸Y
A‡cÿv 4 †ewk| msL¨vwU KZ?14Zg wewmGm
47 36 25 14 DËi: M
25  A¼Ø‡qi mgwó = 2 + 5 = 7, mgwói 3 ¸Y = 73 = 21  25 - 21 = 4 †ewk|
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Math Tutor 19
MATH TUTOR
Written by
Kabial Noor

Math Tutor 19
Aa¨vq 02
ev¯Íe msL¨v (Real Number)
Part 02
01.13 ARvbv msL¨v wbY©q
 †R‡b wbb – 10
 Gai‡bi mgm¨v mgvav‡bi †ÿ‡Î ïiæ‡ZB GKwU msL¨v x a‡i wbb, Zvici `v‡M hv hv †hfv‡e ejv Av‡Q †m Abymv‡i
GwM‡q hvb| A_©vr, †hvM ej‡j †hvM Kiæb, we‡qvM ej‡j we‡qvM Kiæb...| †k‡li w`‡K GKUv P~ovšÍ dj (†hvMdj/
we‡qvMdj/fvMdj/¸Ydj †h‡KvbwU n‡Z cv‡i) †`qv _vK‡e| Gevi Avcbvi a‡i †bqv cÖvß dj mgvb mgvb P~ovšÍ
d‡j wj‡L Zzjbv Kiæb| e¨m&, msL¨vwU P‡j Avm‡e| †hgb-
†Kvb msL¨v n‡Z 175 we‡qvM K‡i 130 †hvM Ki‡j †hvMdj 297 n‡e? ivóªvqË e¨vsK Awdmvi - 97 |
awi, msL¨vwU = x | Pjyb, `v‡M hv hv ejv Av‡Q †mwU AbymiY Kiv hvK- x – 175 + 130| dvBbvwj, Gevi
Zzjbv Kiæb- x – 175 + 130 = 297 x = 342|
 D‡ëv †g_W AbymiY Kiæb: D‡ëv †g_W n‡”Q ÔP~ovšÍ djÕ †_‡K wcwQ‡q wcwQ‡q ïiæi RvqMvq wd‡i Avmv| A‡bKUv
mvg‡bi w`‡K GwM‡q wM‡q bv Ny‡i cybivq Av‡Mi RvqMvq wd‡i Avmv|
mnR K_vq, hvevi mgq mvg‡b cv †d‡j‡Qb, Avmvi mgq wcQ‡b cv †dj‡Z n‡e|
+ 5 – 2  3 = 39  {(GKwU msL¨v + 5) – 2}  3 = 39
D‡ëv †g_‡W Avgiv 39 †_‡K wcwQ‡q wcwQ‡q ÔGKwU msL¨vÕi RvqMvq †cŠQe| GRb¨ 39 Gi Av‡M 3 ¸Y K‡iwQjvg,
GLb 39 †K 3 Øviv fvM Kie = 39  3 = 13| 3 Gi Av‡M 2 we‡qvM K‡iwQjvg GLb 13 Gi mv‡_ 2 †hvM Kie
= 13 + 2 = 15| 2 Gi Av‡M Avgiv 5 †hvM K‡iwQjvg Gevi 15 †_‡K 5 we‡qvM Kie = 15 – 5 = 10| e¨m&,
Avgiv ÔGKwU msL¨vÕi RvqMvq wd‡i Avmjvg|
 civgk©: D‡ëv †g_WwU P~ovšÍ dj †_‡K ïiæ Ki‡Z n‡e, Zvici ch©vµ‡g †h †h wPý _vK‡e Zvi wecixZ wP‡ýi KvR
Ki‡Z n‡e| †hvM _vK‡j we‡qvM, we‡qvM _vK‡j †hvM, ¸Y _vK‡j fvM, fvM _vK‡j ¸Y Ki‡Z n‡e|
(K) mgxKiY I D‡ëv †g_W eënvi K‡i mgvavb
 hLb GKwU msL¨vi mv‡_ †hvM, we‡qvM, ¸Y, fvM avivevwnKfv‡e GK wbtk¦v‡m eënvi K‡i P~ovšÍ d‡j †cuŠQv nq ïay
ZLbB D‡ëv †g_W eënvi Kiv hvq| G av‡ci mgxKiY¸‡jvi w`‡K jÿ¨ K‡i †`Lyb, cÖwZwU mgxKi‡Y x GKeviB
eënvi Kiv n‡q‡Q| A_P (L) av‡ci mgxKiY¸‡jv‡Z x GKvwaKevi eënvi Kiv n‡q‡Q, ZvB (L) av‡c D‡ëv
†g_W eënvi Kiv hv‡e bv|)
105. †Kvb msL¨vi m‡½ 7 †hvM K‡i, †hvMdj‡K 5 w`‡q
¸Y K‡i, ¸Ydj‡K 9 w`‡q fvM K‡i, fvMdj †_‡K
3 we‡qvM Kiv‡Z we‡qvMdj 12 nq| msL¨vwU KZ? eb
I cwi‡ek gš¿Yvj‡qi mnKvix cwiPvjK t 95
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¯’vb
MšÍe¨¯’j
Mr X hvÎvi‡¤¢i ¯’vb †_‡K
hvÎv ïiæ K‡i MšÍe¨¯’‡j
†cŠQj| MšÍe¨¯’j †_‡K bv
Ny‡i D‡ëvfv‡e Avevi
hvÎvi‡¤¢i ¯’v‡b wd‡i Avmj|
GwUB n‡”Q D‡ëv †g_W!
GKwU
msL¨v

20Math Tutor
awi, msL¨vwU x
kZ©g‡Z, 12
3
9
5
7



 )
(x
ev, 15
9
5
7


 )
(x
ev, 5
7 
 )
(x = 135 ev, 7

x =
5
135
ev, 7

x = 27  x = 27 - 7 = 20|
 D‡ëv †g_‡W mgvavb: 12 Gi Av‡M 3 we‡qvM Kiv
n‡q‡Q, GLb 12 Gi mv‡_ 3 †hvM Ki‡Z n‡e = 12
+ 3 = 15| 3 Gi Av‡M 9 Øviv fvM Kiv n‡q‡Q,
GLb 15 Gi mv‡_ 9 ¸Y Ki‡Z n‡e = 15 9 =
135| 9 Gi Av‡M 5 ¸Y Kiv n‡q‡Q, GLb 135 †K
5 Øviv fvM Ki‡Z n‡e = 135  5 = 27| 5 Gi
Av‡M 7 †hvM Kiv n‡q‡Q, GLb 7 we‡qvM Ki‡Z n‡e
= 27 - 7 = 20|
kU©Kv‡U: 12 + 3 = 15  15 9 = 135
 135  5 = 27  27 - 7 = 20| (GB
AvBwWqv e¨envi K‡i Lye `ªæZ mgvavb Kiv hvq)
106. †Kvb msL¨vi A‡a©‡Ki mv‡_ 4 †hvM Ki‡j †hvMdj
nq 14| msL¨vwU KZ? evsjv‡`k e¨vsKAwdmvi -01
10 15
20 25 DËi: M
awi, msL¨vwU = x kZ©g‡Z,
2
x
+ 4 = 14
ev,
2
x
= 10  x = 20|
 D‡ëv †g_‡W mgvavb: 14 Gi Av‡M 4 †hvM Kiv
n‡q‡Q, ZvB 14 †_‡K 4 we‡qvM Ki‡Z n‡e= 14 - 4
= 10| 4 Gi Av‡M A‡a©K _vKvq 10†K wØ¸Y
Ki‡Z n‡e = 102 = 20|
 gy‡L gy‡L: 14-4 = 10,  102 = 20
 civgk©: wb‡Pi mgm¨v¸‡jvi cÖwZwUi mgxKiY •Zwi
K‡i †`qv nj| Avcbvi KvR n‡”Q- mgxKiY †_‡K
x Gi gvb †ei K‡i msL¨vwU wbY©q Kiv|
107. †Kvb msL¨vi
১
৩
mv‡_ 6 †hvM Ki‡j †hvMdj 28 nq|
msL¨vwU KZ? evsjv‡`k e¨vsK Gwm÷¨v›U wW‡i±i -08
44 66
42 84 DËi: L
 28 - 6 = 22  22  3 = 66|
 mgxKiY:
3
x
+ 6 = 28
108. †Kvb msL¨vi GK PZz_v©sk †_‡K 4 we‡qvM Ki‡j
we‡qvMdj nq 20| msL¨vwU KZ? we‡Kwe Awdmvi - 07
12 24
36 96 DËi: N
 20 + 4 = 24  24  4 = 96|
 mgxKiY:
4
x
- 4 = 20
109. †Kvb msL¨vi GK cÂgvsk †_‡K 5 we‡qvM Ki‡j
we‡qvMdj nq 10| msL¨vwU KZ?evsjv‡`kK…wle¨vsKwj.
(wmwbqiAwdmvi)2011
15 25
50 75 DËi: N
 10 + 5 = 15  15  5 = 75|
 mgxKiY:
5
x
- 5 = 10
110. †Kvb msL¨vi wØ¸‡Yi mv‡_ 2 †hvM Ki‡j †hvMdj 88
n‡e? ¯^v¯’¨gš¿Yvj‡qiAaxb†mevcwi`ß‡iiwmwbqi÷vdbvm©:16
41 42
44 43 DËi: N
 88 - 2 = 86  86  2 = 43 (wØ¸Y Gi
wecixZ A‡a©K)
 mgxKiY: 2x + 2 = 88
111. GKwU msL¨vi wØ¸‡Yi mv‡_ 9 †hvM Kiv nj| hw`
cÖvß djvdjwU‡K wZb¸Y Kiv nq Zvn‡j 75 nq|
msL¨vwU KZ? iƒcvjx e¨vsK(wmwbqiAwdmvi)2013
3.5 6
8 †Kv‡bvwUB bq DËi: M
 75 3 = 25  25-9 = 16  16 2 = 8|
 mgxKiY: (2x + 9) 3 = 75
112. GKwU msL¨vi e‡M©i mv‡_ 4 †hvM Ki‡j †hvMdj 40
nq| msL¨vwU KZ? GKwUevwoGKwULvgvi(Dc‡Rjvmgš^qKvix)17
4 5
8 6 DËi: N
 40 - 4 = 36  36 = 6| (eM© Gi
wecixZ eM©g~j)
 mgxKiY: x2
+ 4 = 40
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Math Tutor 21
(L) mgxKiY e¨envi K‡i mgvavb
 †R‡b wbb – 11 (AwZ `ye©j‡`i Rb¨)
 GKwU msL¨v wØ¸Y = 2x, wZb¸Y = 3x, Pvi¸Y = 4x cvuP¸Y = 5x BZ¨vw`|
 GKwU msL¨vi A‡a©K =
2
x
, GK-Z…Zxqvsk =
3
x
, GK-PZy_©vsk =
4
x
, `yB-Z…Zxqvsk =
3
2x
, wZb-cÂgvsk =
5
3x
 GKwU msL¨v I Zvi wecixZ fMœvsk = x I
x
1
, 2 I
2
1
, 7 I
7
1
BZ¨vw`|
113. GKwU msL¨vi wZb¸‡Yi mv‡_ wØ¸Y †hvM Ki‡j 90
n‡e| msL¨vwU KZ? cÖwZiÿvgš¿Yvjqwmwfwjqvb÷vdAwdmviGes
mn:cwi:wb‡qvM:16/cwiKíbvgš¿Yvjqwb‡qvMcixÿv:16
16 18
20 24 DËi: L
awi, msL¨vwU x| kZ©g‡Z, 3x + 2x = 90
ev, 5x = 90  x = 18|
114. †Kvb GKwU msL¨vi 13 ¸Y †_‡K 4 ¸Y ev` w`‡j
171 nq, msL¨vwU KZ? cÖv_wgK I MYwkÿv wefv‡M mnKvix
cwiPvjK -01
15 17
19 29 DËi: M
awi, msL¨vwU x| kZ©g‡Z, 13x - 4x = 171
ev, 9x = 171  x = 19|
115. †Kvb msL¨vi 9 ¸Y †_‡K 15 ¸Y 54 †ewk? AvenvIqv
Awa`ß‡ii mnKvix AvenvIqvwe` -95
9 15
54 6 DËi: K
ev, 6x = 54  x = 9|
116. †Kvb msL¨vi 6 ¸Y n‡Z 15¸Y 63 †ewk? Z_¨
gš¿Yvj‡qi Aax‡b mnKvix cwiPvjK, †MÖW-2t03
6 7
3 9 DËi: L
ev, 9x = 63  x = 7|
117. GKwU msL¨vi A‡a©K Zvi GK Z…Zxqvs‡ki PvB‡Z 17
†ewk| msL¨vwU KZ? cÖwZiÿv gš¿Yv. Aax‡b mvBdvi Awdmvi- 99
52 84
102 204 DËi: M
awi, msL¨vwU x| kZ©g‡Z,
2
x
-
3
x
= 17
ev,
6
2
3 x
x 
= 17 ev, x = 102 |
118. GKwU msL¨v I Zvi wecixZ fMœvs‡ki †hvMdj
msL¨vwUi wØ¸‡Yi mgvb| msL¨vwU KZ? weweG : 94-95
1 -1
1 A_ev -1 2 DËi: M
awi, msL¨vwU x | kZ©g‡Z, x +
x
1
= 2x
ev,
x
1
= x ev, x2
= 1  x =  1
119. GKwU msL¨vi 5 ¸‡Yi mv‡_ Zvi eM© we‡qvM Ki‡j
Ges 6 we‡qvM Ki‡j we†qvMdj k~b¨ nq| msL¨vwU -
13Zg†emiKvixwkÿKwbeÜbIcÖZ¨vqcixÿv(¯‹zj/mgchv©q):16
1 A_ev 2 3 A_ev 4
2 A_ev 3 3 A_ev 4 DËi: M
awi, msL¨vwU x | kZ©g‡Z, 5x - x2
- 6 = 0
ev, x2
- 5x + 6 = 0 ev, x2
- 3x - 2x + 6
ev, x (x - 3) - 2(x -3) = 0 ev, (x - 3) (x -2)
= 0  x = 3 ev x = 2 |
120. †Kvb msL¨vi wØ¸‡Yi mv‡_ 3 †hvM Ki‡j †hvMdj
msL¨vwUi A‡cÿv 7 †ewk nq| msL¨vwU KZ? evsjv‡`k
†c‡Uªvwjqvg G·‡cøv‡ikbGÛ†cÖvWvKkb†Kv¤úvwbwj.(ev‡c·)-17
3 4
5 6 DËi: L
awi, msL¨vwU x | kZ©g‡Z, 2x + 3 = x + 7
ev, x = 4 |
121.GKwU msL¨vi wØ¸‡Yi mv‡_ 20 †hvM K‡i cÖvß
djvdj msL¨vwUi 8 ¸Y †_‡K 4 we‡qvM K‡i cÖvß
djvd‡ji mgvb| msL¨vwU KZ? c~evjxe¨vsKwj.(†UªBwb
A¨vwm‡÷›U)2017
2 3
4 6 DËi: M
awi, msL¨vwU x | kZ©g‡Z, 2x + 20 = 8x - 4
ev, 6x = 24  x = 4 |
122. GKwU msL¨vi 4 ¸‡Yi mv‡_ 10 †hvM Ki‡j DËi nq
msL¨vwUi 5 ¸Y A‡cÿv 5 Kg| msL¨vwU KZ? Bmjvgx
e¨vsKwj.(K¨vk)2017
30 20
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22Math Tutor
25 15 DËi: N
awi, msL¨vwU x | kZ©g‡Z, 4x + 10 = 5x - 5
 x = 15 |
123. 13 Ges GKwU msL¨vi mgwói GK Z…Zxqvsk,
msL¨vwUi wØ¸‡Yi †P‡q 1 †ewk| msL¨vwU †ei Kiæb|
evsjv‡`ke¨vsK (A¨vwm‡÷›UwW‡i±i)2012
6 2
5 3 DËi: L
awi, msL¨vwU x [13 Ges GKwU msL¨vi mgwói GK
Z…Zxqvsk = (13 + x)
3
1
 =
3
13 x

]
kZ©g‡Z,
3
13 x

= 2x + 1 ev, 6x + 3 = 13 + x
ev, 5x = 10  x = 2
01.14 wefvR¨Zv I fvRK msL¨v wbY©q
(K) wefvR¨Zvi bxwZ
 2 Øviv wefvR¨: †Kv‡bv msL¨vi GKK ¯’vbxq A¼wU k~b¨ (0) A_ev †Rvo n‡j cÖ`Ë msL¨vwU 2 Øviv wefvR¨ n‡e|
A_ev msL¨vwUi †k‡l 1wU k~b¨ (0) _vK‡jI 2 Øviv wefvR¨ n‡e|
 g‡b ivLyb: †h msL¨vi †k‡l 1 wU 0, Zv 2 QvovI 5 I 10 Øviv wefvR¨|
†hgb- 24| GLv‡b 4, 2 Øviv wefvR¨| 30 msL¨vwU 2, 5 I 10 Øviv wefvR¨|
 4 Øviv wefvR¨: †Kvb msL¨vi GKK I `kK ¯’vbxq A¼ `ywU Øviv MwVZ msL¨v 4 Øviv wefvR¨ n‡j cÖ`Ë msL¨vwU 4
Øviv wefvR¨ n‡e| A_ev msL¨vwUi †k‡l 2 wU k~b¨ (00) _vK‡jI 4 Øviv wefvR¨ n‡e|
 g‡b ivLyb: †h msL¨vi †k‡l 2 wU k~b¨ (00), Zv 4 QvovI 25 I 100 Øviv wefvR¨|
†hgb- 728| GLv‡b 28, 4 Øviv wefvR¨| 500 msL¨vwU 4, 25 I 100 Øviv wefvR¨|
 8 Øviv wefvR¨: †Kv‡bv msL¨vi GKK, `kK I kZK ¯’vbxq A¼ wZbwU Øviv MwVZ msL¨v 8 Øviv wefvR¨ n‡j cÖ`Ë
msL¨vwU 8 Øviv wefvR¨ n‡e| A_ev msL¨vwUi †k‡l 3wU k~b¨ (000) _vK‡jI 8 Øviv wefvR¨ n‡e|
 g‡b ivLyb: †h msL¨vi †k‡l 3wU k~b¨ (000), Zv 8 QvovI 125 I 1000 Øviv wefvR¨|
†hgb- 7136| GLv‡b †kl wZbwU AsK Øviv MwVZ msL¨v 136, 8 Øviv wefvR¨| 7000 msL¨vwU
8, 125 I 1000 Øviv wefvR¨|
 †KŠkj: 2 (21
) Gi †ÿ‡Î †kl 1 wU A¼, 4 (22
) Gi †ÿ‡Î †kl 2wU A¼ Ges 8 (23
) Gi †ÿ‡Î †kl 3 wU
A¼ fvM Kiv †M‡j cÖ`Ë A¼wU h_vµ‡g 2, 4 I 8 Øviv wefvR¨ n‡e| (cvIqvi †`‡L g‡b ivLyb)
124. wb‡Pi †KvbwU 4 Øviv wefvR¨? evwYR¨gš¿Yvj‡qiAax‡bevsjv‡`kU¨vwidKwgkbwimvm© Awdmvi:10
214133 510056 322569 9522117 DËi: L
125. 91876 * 2 msL¨vwU 8 Øviv wbt‡k‡l wefvR¨ n‡j * Gi RvqMvq †Kvb ÿz`ªZg c~Y©msL¨v e¨envi Kiv hv‡e?
evsjv‡`k e¨vsK A¨vwm‡÷›U wW‡i±i 14
1 2 3 4 DËi: M
†k‡li wZbwU wWwRU (6 * 2) hw` 8 Øviv wefvR¨ nq Zvn‡j cÖ`Ë msL¨vwU 8 Øviv wefvR¨ n‡e| * wP‡ýi RvqMvq
1, 2, 3, 4 Gi gvS †_‡K 3 emv‡j msL¨vwU (632) 8 Øviv wefvR¨ n‡e|
 3 Øviv wefvR¨: †Kv‡bv msL¨vi A¼¸‡jvi †hvMdj 3 Øviv wefvR¨ n‡j, cÖ`Ë msL¨vwU 3 Øviv wefvR¨ n‡e|
†hgb- 126  1 + 2 + 6 = 9  9  3 = 3|  126 msL¨vwU 3 Øviv wefvR¨|
 9 Øviv wefvR¨: †Kv‡bv msL¨vi A¼¸‡jvi †hvMdj 9 Øviv wefvR¨ n‡j, cÖ`Ë msL¨vwU 9 Øviv wefvR¨ n‡e|
†hgb- 1593  1 + 5 + 9 + 3 = 18  18  9 = 2|  1593 msL¨vwU 9 Øviv wefvR¨|
126. wb‡Pi †Kvb msL¨vwU 3 Øviv wb:‡k‡l wefvR¨ bq? cvwbDbœqb†ev‡W©i AwdmmnvqK:15
126 141 324 139 DËi: N
127. 456138 msL¨vwU wb‡¤œi †Kvb msL¨v Øviv wefvR¨?
5 21 9 19 DËi: M
128. 9 w`‡q wefvR¨ 3 A¼ wewkó GKwU msL¨vi cÖ_g A¼ 3, Z…Zxq A¼ 8 n‡j ga¨g A¼wU KZ? kÖgIKg©ms¯’vbgš¿Yvj‡qiAaxb
KjKviLvbvIcÖwZôvbcwi`k©bcwi`ß‡iimnKvixcwi`k©K:05
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Math Tutor 23
6 7 8 9 DËi: L
1g A¼ I 3q A‡¼i †hvMdj = 3 + 8 = 11, hv 9 Øviv wefvR¨ bq| 11 Gi cieZx© 18 msL¨vwU 9 Øviv
wefvR¨, GRb¨ Avgv‡`i 7 †hvM Ki‡Z n‡e| 3 I 8 Gi gv‡S 7 emv‡j msL¨vwU `vuovq 378, hvi A¼¸‡jvi
†hvMdj nq 3 + 7 + 8 = 18| Gevi 18 msL¨vwU 9 Øviv wefvR¨, Zvn‡j 378 msL¨vwUI 9 Øviv wefvR¨|
129. 481 * 673 msL¨vwU 9 Øviv wbt‡k‡l wefvR¨ n‡j, * Gi ¯’v‡b †Kvb ÿz`ªZg c~Y©msL¨v n‡e? AMÖYx e¨vsK wj. (wmwbqi
Awdmvi) 2017
2 7 5 6 DËi: L
 6 Øviv wefvR¨: †Kv‡bv msL¨v‡K 2 Ges 3 Øviv wefvR¨ n‡j msL¨vwU 6 Øviv wefvR¨ n‡e|
 †KŠkj: 6 Øviv wefvR¨ msL¨vwU Aek¨B †Rvo n‡e, ZvB †mwU Aek¨B 2 Øviv wefvR¨ n‡e| Avcbvi KvR
n‡”Q ïay 3 Gi wefvR¨Zv bxwZ cÖ‡qvM K‡i cixÿv K‡i †bqv| 4536
130. 5 * 2 msL¨vwU hw` 6 Øviv wbt‡k‡l wefvR¨ nq, Zvn‡j * ¯’v‡b †Kvb AsKwU em‡e? evsjv‡`k e¨vsK (Awdmvi K¨vk) 16
2 3 6 7 DËi: K
msL¨vwU‡K hw` 2 I 3 Øviv fvM Kiv hvq Zvn‡j GwU 6 Øviv wbt‡k‡l wefvR¨ n‡e| msL¨vwUi †kl AsK †Rvo
_vKvq GwU 2 Øviv wbt‡k‡l wefvR¨| Gevi 3 Gi wefvR¨Zvi bxwZ Abyhvqx 2, 3, 6 I 7 Gi gvS †_‡K Ggb GKwU
AsK 5 * 2 Gi * RvqMvq emv‡Z n‡e †hb AsK¸‡jv †hvM Ki‡j †hvMdj 3 Øviv wefvR¨ nq| G‡ÿ‡Î 2 emv‡j
522 nq, †hLv‡b AsK¸‡jvi mgwó 5 + 2 + 2 = 9, hv 3 Øviv wefvR¨|
 7 Øviv wefvR¨: (1) †Kv‡bv msL¨vi GKK ¯’vbxq AsK‡K 5 ¸Y K‡i Aewkó As‡ki mv‡_ †hvM Ki‡j, †hvMdj
hw` 7 Øviv wefvR¨ nq Z‡e g~j msL¨vwUI 7 Øviv wefvR¨ n‡e| †hgb- 798 79 (85)
 79 + 40 = 119  119  7 = 17| myZivs, 798 msL¨vwU 7 Øviv wefvR¨|
(2) †Kv‡bv msL¨vi GKK ¯’vbxq AsK‡K 2 Øviv ¸Y K‡i Aewkó msL¨v †_‡K we‡qvM Kivi ci
we‡qvMdj 7 w`‡q wefvR¨ n‡j g~j msL¨vwUI 7 w`‡q wefvR¨ n‡e| †hgb- 861
 86 (12)  86 - 2 = 84  84  7 = 12|  msL¨vwU 7 Øviv wefvR¨|
 13 Øviv wefvR¨: (1) †Kv‡bv msL¨vi GKK ¯’vbxq AsK‡K 4 Øviv ¸Y K‡i Aewkó As‡ki mv‡_ †hvM Ki‡j,
†hvMdj hw` 13 Øviv wefvR¨ nq Z‡e g~j msL¨vwUI 13 Øviv wefvR¨ n‡e| †hgb- 14131
 1413(14)  1413 + 4 = 1417  1417  13 = 109|  msL¨vwU 13 Øviv wefvR¨|
 17 Øviv wefvR¨: (1) †Kv‡bv msL¨vi GKK ¯’vbxq AsK‡K 12 Øviv ¸Y K‡i Aewkó As‡ki mv‡_ †hvM Ki‡j,
†hvMdj hw` 17 Øviv wefvR¨ nq Z‡e g~j msL¨vwUI 17 Øviv wefvR¨ n‡e| †hgb- 8738
 873 (812)  873 + 96 = 969  959  17 = 57  msL¨vwU 17 Øviv wefvR¨|
 civgk©: 7 Gi wefvR¨Zvi bxwZwU LyeB ¸iæZ¡c~Y©, ZvB gyL¯’ ivLyb|
131. wb‡Pi †KvbwU 2 Ges 7 Øviv wefvR¨? Bangladesh BankOfficer:01
365 362 361 350 DËi: N
132. wb‡Pi †Kvb msL¨vwU 3 Ges 7 Df‡qi Øviv wbt‡k‡l wefvR¨? K‡›Uªvjvi†Rbv‡ijwW‡dÝdvBb¨vÝ-GiKvh©vj‡qiAaxbRywbqiAwWUi2019
303 341 399 406 DËi: M
 11 Øviv wefvR¨: (1) †Kv‡bv msL¨vi GKK ¯’vbxq AsK Aewkó AsK¸‡jv †_‡K we‡qvMdj 11 Øviv wefvR¨ n‡j
msL¨vwU 11 Øviv wefvR¨ n‡e| †hgb- 1243  124 - 3 = 121  121  11 = 11|
1045  104 - 5 = 99  99  11 = 9|  msL¨v `ywU 11 Øviv wefvR¨|
(2) msL¨vwUi AsK¸‡jv‡K †kl w`K †_‡K †Rvov †Rvov K‡i †hvM Ki‡j †hvMdj 11 Øviv wefvR¨
n‡e| 1243  12 + 43 = 55  55  11 = 5 |  msL¨vwU 11 Øviv wefvR¨|
715  7 + 15 = 22  22  11 = 2|  msL¨vwU 11 Øviv wefvR¨|
(3) †Kv‡bv msL¨vi we‡Rvo ¯’vbxq As‡Ki mgwó Ges †Rvo ¯’vbxq As‡Ki mgwói cv_©K¨ k~Y¨ n‡j
msL¨vwU 11 Øviv wefvR¨ n‡e| †hgb- 1122  (1 + 2) - (1 + 2) = 3 - 3 = 0|
 msL¨vwU 11 Øviv wefvR¨|
 g‡b ivLyb: (3) bs wbqg †Rvo msL¨K A‡¼i †ÿ‡Î mwVK DËi w`‡jI we‡Rvo msL¨K A‡¼i †ÿ‡Î A‡bK mgq
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24Math Tutor
mwVK DËi †`q bv| †hgb- 209, 726, 759 BZ¨vw` 11 Øviv wefvR¨ n‡jI (3) bs wbqgvbyhvqx cÖgvY
Ki‡Z mÿg n‡eb bv|
133. wb‡Pi †Kvb msL¨vwU 11 Øviv wbt‡k‡l wefvR¨? c~evjx e¨vsK wj. †UªBwb A¨vwm‡÷›U †Ujvi) 2017
235641 245642 315624 415624 DËi: N
(L) fvRK msL¨v
 †R‡b wbb – 12
 fvRK : †h †h ivwk Øviv †Kvb msL¨v‡K fvM Kiv hvq, †m †m ivwk H msL¨vi fvRK| †hgb- 20 †K 1, 2, 4, 5,
10, 20 Øviv fvM Kiv hvq, ZvB 1, 2, 4, 5, 10, 20 n‡”Q 20 Gi fvRK|
 fvR¨ : fvRK Øviv †h msL¨v‡K fvM Kiv hvq, H msL¨v‡K fvR¨ e‡j| †hgb- Dc‡ii D`vni‡Y 20 n‡”Q fvR¨|
 g‡b ivLyb- 1 †h‡Kvb msL¨vi fvRK, KviY 1 Øviv mKj msL¨v wefvR¨| fvRK/Drcv`K/¸YbxqK GKB wRwbm|
fvR¨/¸wYZK GKB wRwbm|
 fvRK msL¨v wbY©‡qi mvaviY wbqgt 32 Gi fvRK msL¨v wbY©q Kiv hvK| 32 Gi fvRK mg~n n‡”Q 32 †K †h †h
msL¨v Øviv fvM Kiv hvq| 24 †K 1, 2, 3, 4, 6, 8, 12 I 24 Øviv fvM Kiv hvq| A_©vr, 24 Gi fvRKmg~n = 1,
2, 3, 4, 6, 8, 12 I 24|  24 Gi fvRKmsL¨v n‡”Q 8 wU| GB c×wZ‡Z eo msL¨vi fvRK msL¨v wbY©q Kiv
KwVb I mgqmv‡cÿ, ZvB fvRK msL¨v wbY©‡q Avgiv kU©KvU wbqg AbymiY Kie|
 fvRK msL¨v wbY©‡qi kU©KvU wbqgt cÖ_‡g †h msL¨vi fvRK msL¨v wbY©q Kie, †m msL¨vwU‡K †gŠwjK Drcv`‡K
we‡kølY Kie|
2 24 24 Gi †gŠwjK Drcv`Kmg~n = 2  2  2  3|
2 12 GLv‡b Drcv`Kmg~‡ni gv‡S 2 Av‡Q 3wU Ges 3 Av‡Q 1wU| GLb m~P‡Ki wbqgvbyhvqx 2 Gi cvIqvi 3 Ges
2 6 3 Gi cvIqvi 1 wjLyb Gfv‡e- 23
 31
| Zvici wfwË 2 I 3 †K †Ku‡U w`b- 23
 31
| Gevi wfwË ev`
3 w`‡q cÖwZwU cvIqvi Gi mv‡_ 1 K‡i †hvM Kivi ci ¸Y Kiæb- (3 + 1)  (1 + 1) = 4  2
= 8| e¨m&, GLv‡b cÖvß 8 n‡”Q 24 Gi †gvU fvRK msL¨v|
134. 36 msL¨vwUi †gvU KZ¸‡jv fvRK i‡q‡Q? cjøxDbœqb
†ev‡W©iwnmvemnKvix:14
6wU 8wU
9wU 10wU DËi: M
36 Gi †gŠwjK Drcv`Kmgg~n = 2233
= 22
 32
 fvRK msL¨v = 22
 32
= (2 + 1)  (2 + 1) = 3  3 = 9 wU |
135. 72 Gi fvRK msL¨v KZ? 26ZgwewmGm
7 8
12 13 DËi: M
72Gi †gŠwjK Drcv`Kmg~n = 22233
= 23
 32
 fvRK msL¨v = 23
 32
= (3+1)  (2+1) = 43 = 12 wU|
136. 540 msL¨vwUi KZ¸‡jv fvRK Av‡Q? AvenvIqvAwa`ß‡ii
mnKvixAvenvIqvwe`:04]
18 20
22 24 DËi: N
540 Gi †gŠwjK Drcv`Kmg~n = 2233
35 = 22
33
51
fvRK msL¨v = 22
33
51
= (2+1)(3+1)(1+1) = 342 = 24 wU|
137. 1008 msL¨vwUi KqwU fvRK Av‡Q? Dc‡RjvI_vbvwkÿv
Awdmvit05/_vbvwbev©PbAwdmvit04
20 24
28 30 DËi: N
1008 Gi †gŠwjK Drcv`Kmg~n = 2222
337 = 24
 32
71
fvRK msL¨v = 24
32
71
= (4+1)(2+1)
(1+1) = 532= 30wU|
138. wb‡Pi †Kvb c~Y© msL¨vwU mev©waK msL¨K fvRK Av‡Q?
29ZgwewmGm
88 91
95 99 DËi: K
KvQvKvwQ msL¨vi gv‡S †Rvo msL¨vi fvRK msL¨v
memgq †ewk _v‡K| GLv‡b 88 Gi fvRK msL¨v †ewk|
139. 32 Ges 64 Gi fvRK msL¨vi cv_©K¨ KZ? IBA:88-89
3 2
1 †Kv‡bvwUB bq DËi: M
32 Gi fvRK msL¨v 6wU Ges 64 Gi fvRK msL¨v
7wU|  fvRK msL¨vi cv_©K¨ = 7 - 6 = 1|
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Math Tutor 25
140. wb¤œwjwLZ msL¨v¸‡jvi g‡a¨ †KvbwUi fvRK msL¨v
†e‡Rvo? 16ZgwewmGm
2048 1024
512 48 DËi: L
c~Y©eM© msL¨vi fvRK msL¨v memgq †e‡Rvo nq|
cÖ`Ë Ackb¸‡jvi gv‡S 1024 n‡”Q c~Y©eM© msL¨v,
1024 Gi fvRK msL¨v †e‡Rvo|
(M) x I y Gi gv‡S ---- Øviv wefvR¨ fvRK msL¨v wbY©q
 †R‡b wbb – 13
 1 †_‡K 25 Gi gv‡S 5 Øviv wefvR¨ msL¨v KqwU? GLv‡b 5 Øviv wefvR¨ ej‡Z eySv‡”Q 1 †_‡K 25 Gi gv‡S
GiKg KqwU msL¨v Av‡Q hv‡`i‡K 5 Øviv fvM Kiv hvq| GiKg msL¨vmg~n n‡”Q 5, 10, 15, 20 I 25 | 1 †_‡K
25 Gi gv‡S GB 5wU msL¨v Av‡Q hv‡`i‡K 5 Øviv fvM Kiv hvq| Zvn‡j cÖ`Ë cÖ‡kœi DËi n‡”Q 5wU| gRvi welq
n‡”Q- 25 †K 5 Øviv fvM Ki‡j Avgiv mivmwi GB GKB DËi 5 †c‡q hvB| A_©vr, G ai‡Yi mgm¨vi mgvavb fvM
K‡i KivB me‡P‡q mnR - 25  5 = 5|
 g‡b ivLyb- 5 Øviv wefvR¨ msL¨vmg~ni w`‡K jÿ¨ Kiæb- cÖwZwU msL¨vB 5 Gi ¸wYZK| Zvi gv‡b 1 †_‡K 25 Gi
gv‡S 5 Øviv wefvR¨ msL¨v KqwU? GB K_vwUi Av‡iv GKwU A_© Av‡Q, †mwU n‡”Q- 1 †_‡K 25 Gi gv‡S 5 Gi
¸wYZK KqwU?
141. 1 †_‡K 80 ch©šÍ 4 Øviv wefvR¨ msL¨v KqwU?
19 20 21 22 DËi: L
1 †_‡K 80 ch©šÍ 4 Gi ¸wYZK/ 4 Øviv wefvR¨ msL¨v = 80  4 = 20 wU|
142. 12 I 96 Gi g‡a¨ (GB `ywU msL¨vmn) KqwU msL¨v 4 Øviv wefvR¨? evsjv‡`kcjøxwe`y¨Zvqb†ev‡W©imnKvixmwPe/mnKvixcwiPvjK
(cÖkvmb):16;cÖvK-cÖv_wgKmnKvixwkÿK:14;18ZgwewmGm
21 23 24 22 DËi: N
cÖ‡kœ 12 †_‡K 96 ch©šÍ 4 Gi KqwU ¸wYZK †mwU †ei Ki‡Z ejv n‡q‡Q| Avgiv hw` welqwU †f‡½ †f‡½ †`wL-
1 †_‡K 96 ch©šÍ 4 Gi ¸wYZK/ 4 Øviv wefvR¨ msL¨vmg~n = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48,
52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96 | †gvU 24wU| wKš‘ cÖ‡kœ 4 Gi ¸wYZK 12 †_‡K ïiæ n‡Z
n‡e ejv n‡q‡Q, ZvB 4 Gi cÖ_g 2wU ¸wYZK 4 I 8 ev` w`‡Z n‡e| Zvn‡j 12 †_‡K 96 ch©šÍ 4 Øviv wefvR¨
msL¨v/ 4 Gi ¸wYZK n‡e 22 wU| GB mgm¨vwU fvM c×wZ mgvavb Kiv hvK| cÖ_‡g 96 †K 4 Øviv fvM Kiv hvK-
96  4 = 24wU| eyS‡Z cvi‡Qb †Zv? GB 24 wKš‘ G‡m‡Q 1 †_‡K 96 ch©šÍ Gwiqvi Rb¨ | wKš‘ cÖ‡kœ ejv n‡q‡Q
4 Gi ¸wYZK ïiæ n‡e 12 †_‡K| GRb¨ g‡b g‡b wn‡me K‡i 12 Gi Av‡Mi `ywU ¸wYZK 4 I 8 †K ev` w`‡Z
n‡e| A_©vr, fvRK msL¨v = 24 - 2 = 22 wU|
143. 5 I 95 Gi g‡a¨ 5 I 3 Øviv wefvR¨ msL¨v KZwU? cjøx mÂq e¨vsK (K¨vk) 2018; ivóªvqË¡ e¨vsK (wmwbqi Awdmvi) 1998
6wU 9wU 7wU 15wU DËi: K
5 I 95 Gi g‡a¨ 5 I 3 Øviv wefvR¨ msL¨vmg~n- 15, 30, 45, 60, 75, 90| †`Lv hv‡”Q 5 I 3 Øviv wefvR¨
cÖ_g msL¨v 15, hv 5 I 3 Gi j.mv.¸| evKx msL¨v¸‡jv 5 I 3 Gi j.mv.¸Õi ¸wYZK| GRb¨ GKvwaK msL¨v Øviv
wefvR¨ msL¨v PvIqv n‡j H GKvwaK msL¨vi j.mv.¸ †ei K‡i †mwU Øviv fvM Ki‡Z n‡e| †hgb- 5 I 3 Gi
j.mv.¸ 15  95  15 = 6.33 (DËi `kwgK Qvov wb‡Z n‡e) fvRK msL¨v = 6 wU|
(N) KZ †hvM ev we‡qvM Ki‡j wbt‡k‡l wefvR¨ n‡e
144. 1056 Gi mv‡_ me©wb¤œ KZ †hvM Ki‡j †hvMdj 23
Øviv wb:‡k‡l wefvR¨ n‡e? evsjv‡`kK…wlDbœqbK‡c©v‡ik‡bi
mnKvixcÖkvmwbKKg©KZ©v:17
2 3
18 21 DËi: K
1056 †K 23 Øviv fvM K‡i cvB,
23) 1056 ( 45
92
136
115
21
 cÖ`Ë msL¨vi mv‡_ (23 - 21) = 2 †hvM Ki‡j
cÖvß msL¨vwU 23 Øviv wefvR¨ n‡e|
 g‡b ivLyb: cÖ‡kœ †hvM ej‡j ÔfvRK I fvM‡klÕ Gi
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26Math Tutor
cv_©K¨‡K †hvM Ki‡Z nq|
145. 4456 Gi mv‡_ †Kvb ÿz`ªZg msL¨v †hvM Ki‡j
†hvMdj 6 Øviv wb:‡k‡l wefvR¨ n‡e? EXIMBankLtd.
Officer :13
2 3
4 5 DËi: K
146. 105 Gi m‡½ KZ †hvM Ki‡j †hvMdjwU 23 Øviv
wbt‡k‡l wefvR¨ n‡e? cyevjx e¨vsK wj. (Rywbqi
Awdmvi) 2013
3 18
21 10 DËi: N
147. mvZ A‡¼i e„nËg msL¨v wbY©q Kiæb hv 6 Øviv
wefvR¨|
6) 9999999 ( 1666666
9999996
3
fvM‡kl 3 ev` w`‡j cÖvß 9999996 msL¨vwU
wbt‡k‡l wefvR¨|
 cÖkœwU hw` Gfv‡e ejv nZ- mvZ A‡¼i e„nËg msL¨v
†_‡K †Kvb ÿz`ªZg msL¨v we‡qvM w`‡j we‡qvMdj 6
Øviv wbt‡k‡l wefvR¨? - G‡ÿ‡Î DËi nZ 3| we‡qvM
ejv _vK‡j fvM‡klB DËi nq|
148. cuvP A‡¼i ÿz`ªZg †Kvb msL¨v 41 Øviv wb:‡k‡l
wefvR¨? JanataBankLtd.Asst.ExecutiveOff.:(Teller):15
10004 10025
10041 10045 DËi: K
41) 10000 ( 243
82
180
164
160
123
37
cÖ`Ë msL¨vi mv‡_ (41 - 37) = 4 †hvM Ki‡Z n‡e-
10000 + 4 = 10004 | AZGe, cvuP A‡¼
ÿz`ªZg msL¨v 10004, 41 Øviv wbt‡k‡l wefvR¨|
 †R‡b wbb – 14
†hvM ev we‡qvM ejv bv _vK‡j ÿz`ªZg ev e„nËg
msL¨v‡K wbt‡k‡l wefvR¨ Ki‡Z KLb ÔfvM‡klÕ
we‡qvM Ki‡eb A_ev KLb ÔfvRK I fvM‡klÕ Gi
cv_©K¨‡K †hvM Ki‡eb?
 Reve: GwU m¤ú~Y© wbf©i K‡i cÖ`Ë msL¨vi Dci| †hgb
- 24 bs mgm¨vi †ÿ‡Î fvM‡kl 3 we‡qvM K‡iwQ KviY
ÔfvRK I fvM‡klÕGi cv_©K¨ †hvM Ki‡j msL¨vwU
`uvovZ- 9999999 + ( 6- 3) = 10000002,
Zvn‡j ZLb msL¨vwU Avi mvZ A‡¼i e„nËg msL¨v
_vKZ bv| wKš‘ hLb ÔfvM‡klÕ we‡qvM K‡iwQ ZLb
cÖvß 9999996 msL¨vwU mvZ A‡¼I e„nËg msL¨v
wn‡m‡e wU‡K †M‡Q|
Avevi, 25 bs mgm¨v †ÿ‡Î ÔfvRK I fvM‡klÕ Gi
cv_©K¨‡K †hvM K‡iwQ, KviY ÔfvM‡klÕ we‡qvM Ki‡j
msL¨vwU `uvovZ- 9963, Zvn‡j ZLb msL¨vwU Avi
cvuP A‡¼i ÿz`ªZg msL¨v _vKZ bv| wKš‘ hLb
ÔfvRK I fvM‡klÕ Gi cv_©K¨ †hvM K‡iwQ ZLb cÖvß
10004 msL¨vwU cvuP A‡¼i ÿz`ªZg msL¨v wn‡m‡e
wU‡K †M‡Q|
 g‡b ivLyb: wefvR¨Zvi cÖ‡kœ †hvM ev we‡qvM ejv bv
_vK‡j GKwU kU©KvU g‡b ivLyb- e„nËg msL¨vi †ÿ‡Î
ÔfvM‡kl we‡qvM Ki‡Z nq Ges ÿz`ªZg msL¨vi †ÿ‡Î
ÔfvRK I fvM‡klÕGi cv_©K¨‡K †hvM Ki‡Z nq|
149. GKwU msL¨v‡K 45 w`‡q fvM Ki‡j fvM‡kl 23
_v‡K| hw` H msL¨vwU‡K 9 w`‡q fvM Kiv nq Z‡e
fvM‡kl KZ n‡e? cÖavbgš¿xi Kvh©vjq : IqvPvi
Kb‡÷ej: 2019
3 4
5 100 DËi: M
45 Øviv †h msL¨v‡K fvM Kiv hvq 9 ØvivI H
msL¨v‡K fvM Kiv hvq| Avgiv cÖ‡kœ †`L‡Z cvw”Q 45
Øviv GKwU msL¨v‡K fvM Kivq fvM‡kl 23 Av‡Q|
Avgiv hw` fvM hvIqv AskUzKz x awi, Zvn‡j
fvM‡klmn msL¨vwU n‡e- x + 23| Avgiv Gevi
msL¨vwU‡K 9 Øviv fvM Kie-
9
23

x
=
9
23
9

x
45 †h‡nZz 9 Gi ¸wYZK, †m‡nZz x AskUzKz 45 Øviv
†hgb fvM hv‡e GKBfv‡e 9 ØvivI fvM hv‡e| evKx
_vKj 23 †K 9 Øviv fvM Kiv| Pjyb fvM Kiv hvK-
9 ) 23 ( 2
18
5
A_©vr, H msL¨vwU‡K 9 w`‡q fvM Ki‡j 5 fvM‡kl
_vK‡e|
 GKevi eyS‡Z cvi‡j A¼wU †`Lv gvÎ †m‡K‡ÛB
mgvavb Ki‡Z cvi‡eb|
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Math Tutor 27
01.15 †gŠwjK msL¨v
 †R‡b wbb – 15
 †gŠwjK msL¨v: †h msL¨vi †Kvb cÖK…Z Drcv`K †bB Zv‡K †gŠwjK msL¨v e‡j| A_ev †h msL¨v‡K 1 I H msL¨v
e¨ZxZ Ab¨ †Kvb msL¨v Øviv fvM Kiv hvq bv, Zv‡K †gŠwjK msL¨v e‡j| †hgb- 2, 3, 5, 7 BZ¨vw`|
2, 3, 5 I 7 G 1 I Zviv wb‡Riv e¨ZxZ Ab¨ †Kvb Drcv`K †bB, ZvB Giv †gŠwjK msL¨v|
 †hŠwMK msL¨v: †h msL¨vq 1 I H msL¨v e¨ZxZ Av‡iv Ab¨ †Kvb Drcv`K _v‡K Zv‡K †hŠwMK msL¨v e‡j| †hgb- 4|
4 Gi Dcrcv`Kmg~n- 1, 2, 4| A_©vr, 4-G 1 I 4 QvovI Av‡iv GKwU Drcv`K 2 Av‡Q, ZvB 4 n‡”Q †hŠwMK
msL¨v|
 †gŠwjK msL¨v m¤úwK©Z wKQz ¸iæZ¡c~Y© Z_¨:
(K) 2 e¨ZxZ me †Rvo msL¨v †hŠwMK msL¨v| 2 -B GKgvÎ †Rvo †gŠwjK msL¨v I †QvU †gŠwjK msL¨v|
(L) †gŠwjK wØ‡RvU ev †Rvo †gŠwjK: `ywU †gŠwjK msL¨vi AšÍi 2 n‡j, Zv‡ì †gŠwjK wØ‡RvU e‡j| †hgb- 5, 7|
(M) †gŠwjK wÎ‡RvU: wZbwU †gŠwjK msL¨vi µwgK AšÍi 2 n‡j, Zv‡ì †gŠwjK wÎ‡RvU e‡j| †hgb- 3, 5, 7|
(N) 1 †_‡K 100 ch©šÍ †gŠwjK msL¨v 25wU Ges G‡ì †hvMdj 1060| 101 †_‡K 200 ch©šÍ †gŠwjK msL¨v 21wU|
1 †_‡K 500 ch©šÍ †gŠwjK msL¨v 95wU| 1 †_‡K 1000 ch©šÍ †gŠwjK msL¨v 168wU| 1 †_‡K 5000 ch©šÍ
†gŠwjK msL¨v 669 wU|
150. me‡P‡q †QvU †gŠwjK msL¨v †KvbwU? PubaliBankLtd.(SeniorOfficer) 2017
0 1 2 3 DËi: M
(K) †gŠwjK msL¨vi ZvwjKv
 1 †_‡K 100 ch©šÍ †gŠwjK msL¨vi QK:
cÖ`Ë msL¨v †gŠwjK msL¨v me©‡gvU
1-10 ch©šÍ 2, 3, 5, 7 4 wU
1  50 ch©šÍ 15 wU
11-20 ch©šÍ 11, 13, 17, 19 4 wU
21-30 ch©šÍ 23, 29 2 wU
31- 40 ch©šÍ 31, 37 2 wU
41-50 ch©šÍ 41, 43, 47 3 wU
51-60 ch©šÍ 53, 59 2 wU
51  100 ch©šÍ 10 wU
61-70 ch©šÍ 61, 67 2 wU
71-80 ch©šÍ 71, 73, 79 3 wU
81-90 ch©šÍ 83, 89 2 wU
91-100 ch©šÍ 97 1 wU
1  100 ch©šÍ 25 wU
 g‡b ivLyb : 44 22 3 22 3 21
 101 †_‡K 200 ch©šÍ †gŠwjK msL¨vi QK:
cÖ`Ë msL¨v †gŠwjK msL¨v me©‡gvU
101-110 ch©šÍ 101, 103, 107, 109 4 wU
101  150 ch©šÍ 10 wU
111-120 ch©šÍ 113 1 wU
121-130 ch©šÍ 127 1 wU
131-140 ch©šÍ 131, 137, 139 3 wU
141-150 ch©šÍ 149 1 wU
151-160 ch©šÍ 151, 157 2 wU
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28Math Tutor
161-170 ch©šÍ 163, 167 2 wU
151  200 ch©šÍ 11 wU
171-180 ch©šÍ 173, 179 2wU
181-190 ch©šÍ 181 1 wU
191-200 ch©šÍ 191, 193, 197, 199 4 wU
101  200 ch©šÍ 21 wU
 g‡b ivLyb : 41 1 31 22 21 4
151. wb‡Pi †Kvb msL¨vwU †gŠwjK? ivóªvqË¡ e¨vsKwmwbqiAwdmvi:00
49 51 53 55 DËi: M
(L) †gŠwjK msL¨v wbY©q
 †R‡b wbb – 16 (†gŠwjK msL¨v wbY©‡qi †KŠkj)
 †gŠwjK msLv mn‡R wbY©‡qi Dcvq n‡”Q- 7, 11, 13, 17 N‡ii bvgZv m¤ú‡K© Lye fv‡jv Avq‡Ë¡ ivLv| cvkvcvwk 3,
7, 11 Gi wefvR¨Zvi bxwZI fv‡jvfv‡e AvqË¡ ivLv PvB|
 2 I 5 Gi wefvR¨Zvi bxwZi e¨vcv‡i aviYv: 2 e¨ZxZ †Kvb †Rvo msL¨v †gŠwjK msL¨v nq bv, ZvB memgq †e‡Rvo
msL¨vi gv‡S †gŠwjK msL¨v LyuR‡Z n‡e, GRb¨ †gŠwjK msL¨v †ei Ki‡Z KL‡bv 2 Gi wefvR¨Zvi bxwZi ìKvi n‡e
bv| Gevi †e‡Rvo msL¨vi gv‡S 5 LyeB ¸iæZ¡c~Y©| 5 †gŠwjK msL¨v, wKš‘ evKx †h‡Kvb msL¨vi GKK ¯’v‡b 5 _vK‡j
†mwU †hŠwMK msL¨v| †hgb- 15, 55, 75, 105 BZ¨vw`| GRb¨ 5 Gi wefvR¨Zvi bxwZ wb‡qI gv_v Nvgv‡Z n‡e bv,
KviY †Kvb msL¨vi GKK ¯’v‡b 5 †`L‡jB eySv hv‡e GwU †hŠwMK msL¨v|
 3 Gi wefvR¨Zvi bxwZi e¨vcv‡i aviYv: 3 Gi wefvR¨Zvi bxwZwU fv‡jvfv‡e AvqË¡ ivLv PvB, KviY 3 Gi
wefvR¨Zvi bxwZ w`‡q A‡bK †hŠwMK msL¨v Lye mn‡RB †ei Kiv hvq| †Kvb msL¨v †gŠwjK wKbv, †mwU wbY©‡qi
avç‡jv‡Z cÖ‡e‡ki ïiæ‡ZB 3 Gi wefvR¨Zvi bxwZwU cÖ_‡g cÖ‡qvM K‡i †`L‡eb|
 †gŠwjK msL¨v wbY©‡qi avcmg~n:
 avc-01: cÖ_‡g 3 Gi wefvR¨Zvi bxwZ w`‡q hvQvB Ki‡eb, msL¨vwU †hvwMK wKbv? hw` †hŠwMK nq, Zvn‡j Avi
G‡Mv‡bvi ìKvi †bB| †hgb- 117 msL¨vwU †gŠwjK wKbv? 3 Gi wefvR¨Zvi bxwZ Abyhvqx hvQvB Kiv hvK- 1 + 1 + 7
= 9, †h‡nZz †hvMdj 9, 3 Øviv wefvR¨, †m‡nZz 117 msL¨vwUI 3 Øviv wefvR¨ A_©vr, 117 msL¨vwU †hŠwMK| ZvB Avi 2q
av‡c hvIqvi ìKvi †bB| Gevi Av‡iv GKwU msL¨v †bqv hvK- 143 msL¨vwU †gŠwjK wKbv? cÖ_‡g 3 Gi wefvR¨Zvi bxwZ
Abyhvqx †`Lv hvK- 1 + 4 + 3 = 8, hv 3 Øviv wefvR¨ bq| A_©vr, 143 msL¨vwU 3 Øviv wefvR¨ bq | mveavb! Zvi gv‡b
143 †gŠwjK msL¨v bq| fv‡jv K‡i g‡b ivLyb, 3 Øviv fvM bv †M‡j msL¨vwU‡K wØZxq av‡c wb‡q †h‡Z n‡e|
 avc-02: 2q av‡c cÖ`Ë msL¨vwUi KvQvKvwQ GKwU eM©g~j wb‡Z n‡e Ges D³ eM©g~‡ji c~‡e© †gŠwjK msL¨v †ei
Ki‡Z n‡e| 143
2,3, 5, 7, 11  12
2, 3, 5 †jLvi ìKvi †bB, KviY 2q av‡c Avmvi Av‡MB Avcwb 2, 3, 5 hvQvB K‡i wb‡q‡Qb| Gevi 7 I 11 Gi
†KvbwU Øviv hw` 143 †K fvM Kiv hvq, Zvn‡j msL¨vwU †hŠwMK Avi hw` fvM Kiv bv hvq Zvn‡j msL¨vwU †gŠwjK|
143 †K 11 Øviv fvM Kiv hvq, ZvB 143 †gŠwjK msL¨v bq|
 PP©v Kiæb: 133, 127, 119, 141 |
152. wb‡Pi †KvbwU †gŠwjK msL¨v? 30ZgwewmGm
91 87 63 59 DËi: N
3 Gi wefvR¨Zvi bxwZ Abyhvqx I ev`| 91 = 7  13 Abyhvqx ev`|
153. wb‡Pi †Kvb msL¨vwU †gŠwjK? 10gwewmGm
91 143 47 87 DËi: M
91 I 143 c~‡e© cÖgvY Kiv n‡q‡Q| 3 Gi wefvR¨Zvi bxwZ Abyhvqx ev`| †QvU †QvU msL¨v _vK‡j mivmwi
DËi Kiv hvq|
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Math Tutor 29
154. wb‡Pi †KvbwU †gŠwjK? ivóªvqËe¨vsKwmwbqiAwdmvi:00
49 51 53 55 DËi: M
155. †KvbwU †gŠwjK msL¨v bq? cvewjKmvwf©mKwgk‡bmnKvixcwiPvjK:04
221 227 223 229 DËi: K
cÖ_g av‡c 3 Gi wefvR¨Zvi bxwZ Abyhvqx GKwU‡KI ev` †`qv hv‡”Q bv| Gevi wØZxq av‡c hvIqv hvK- cÖ`Ë
me KqwU msL¨v KvQvKvwQ nIqvq Avgvi me KqwUi Rb¨ GKwU eM©g~j wb‡Z cvwi| KvQvKvwQ eM©g~j 15 †bqv hvK|
221 227 223 229
7, 11, 13  15 7, 11, 13  15 7, 11, 13  15 7, 11, 13  15
(221 = 1317) (227, 223, 229 Gi †KvbwUB‡K 7, 11, 13 Øviv fvM Kiv hvq bv)
 221 †gŠwjK msL¨v bq|
 †gŠwjK msL¨v wbY©‡qi †ÿ‡Î †Kvb †UKwb‡Ki `iKvi n‡e bv, hw` Avcwb bvgZv ev wefvR¨Zvq `ÿ _v‡Kb|
(M) x †_‡K y ch©šÍ †gŠwjK msL¨v wbY©q
 †R‡b wbb – 17
 x †_‡K y ch©šÍ fvRK msL¨v wbY©q Ki‡Z wM‡q Avgiv cÖvq fvlvMZ RwUjZvq c‡i hvB| Pjyb fvlvMZ RwUjZv `~i
Kiv hvK-
2 †_‡K 31 ch©šÍ †gŠwjK msL¨v = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 = 11 wU|
(+) (+) †_‡K ch©šÍ
 g‡b ivLyb- †Kvb msL¨v Ô†_‡KÕ gv‡b H msL¨vwU starting point, ZvB H msL¨vmn †gŠwjK msL¨v MYbv Ki‡Z n‡e
Ges †Kvb msL¨v Ôch©šÍÕ gv‡b H msL¨vwU ending point, ZvB Ôch©šÍÕ _vK‡j H msL¨vmn †gŠwjK msL¨v MYbv
Ki‡Z n‡e| A_©vr, Ô‡_‡KÕ I Ôch©šÍÕ _vK‡j starting I ending point mn wn‡me Ki‡Z n‡e|
2 †_‡K 31 Gi g‡a¨ †gŠwjK msL¨v = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 = 10 wU|
(+) (-) †_‡K  Gi g‡a¨
 g‡b ivLyb- Ô‡_‡KÕ _vK‡j H msL¨vmn Ges †Kvb msL¨vi Ôg‡a¨Õ _vK‡j H msL¨v e¨ZxZ wn‡me Ki‡Z nq| Wv‡bi
Q‡K †`Lyb, 31 Gi g‡a¨ gv‡b 31 bq Zvi Av‡Mi msL¨v¸‡jv‡K wb‡`©k Ki‡Q| A_©vr, Ô‡_‡KÕ I Ôg‡a¨Õ _vK‡j
cÖ_gUv wn‡me Ki‡Z n‡e wKš‘ †k‡liUv MYbvq Avm‡e bv|
2 Ges 31 Gi g‡a¨ †gŠwjK msL¨v = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 = 9 wU|
(-) (-)  I/Ges  Gi g‡a¨
 g‡b ivLyb- ÔGesÕ I ÔGi g‡a¨Õ _vK‡j ÔïiæÕ I Ô‡klÕ ev` hv‡e| Q‡K †`Lyb, 2 I 31 ev‡` Zv‡`i g‡a¨ Ae¯’vbiZ
msL¨v¸‡jvi gv‡S †gŠwjK msL¨v †ei Ki‡Z ejv n‡q‡Q|
156. 1 †_‡K 10 ch©šÍ msL¨vi g‡a¨ †gŠwjK msL¨v KZwU?
BankersSelectionCommittee(SeniorOfficer)2018; 10g
wewmGm
4 3
6 5 DËi: K
4wU : 2, 3, 5, 7 |
157. 1 †_‡K 31 ch©šÍ KqwU †gŠwjK msL¨v Av‡Q? wd‡gj
†m‡KÛvix GwmmU¨v›UAwdmvi:99
10 wU 11wU
12 wU 13 wU DËi: L
11wU : 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31
158. 2 Ges 32 -Gi g‡a¨ †gŠwjK msL¨v KqwU? 24Zg
wewmGm
11wU 9wU
8wU 10wU DËi: N
ÔGesÕ I Ôg‡a¨Õ _vKvq 2 I 32 ev‡` wn‡me Ki‡Z
n‡e- 3, 5,7, 11, 13, 17, 19, 23, 29, 31|
159. 10 I 30 Gi g‡a¨ KqwU †gŠwjK msL¨v Av‡Q?
gv`K`ªe¨wbqš¿YAwa`ß‡iimnKvixcwiPvjK:99
4wU 6wU
9wU 5wU DËi: L
6wU : 11,13,17,19,23, Ges 29|
160. 50 -Gi †P‡q †QvU KqwU †gŠwjK msL¨v Av‡Q?
Dc‡Rjv I _vbv wkÿv Awdmvi: 05
mgvavb
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30Math Tutor
10wU 12wU
14 wU 15wU DËi: N
15wU: 2, 3 , 5, 7, 11, 13, 17, 19, 23, 29,
31, 37 , 41, 43 Ges 47|
161. 20 Gi †P‡q eo Ges 200 Gi †P‡q †QvU KZ¸wj
†gŠwjK msL¨v Av‡Q? AvenvIqv Awa`ß‡ii mnKvix
AvenvIqvwe`: 04
35 37
38 40 DËi: M
1 †_‡K 200 ch©šÍ †gŠwjK msL¨v 46 wU Ges 1
†_‡K 20 Gi g‡a¨ †gŠwjK msL¨v 8 wU| GLb 46
†_‡K 8 ev` w`‡j _v‡K 38| myZivs 20 - 200 ch©šÍ
†gŠwjK msL¨v 38 wU|
162. 22 Ges 72 Gi g‡a¨ KqwU †gŠwjK msL¨v i‡q‡Q?
cwiKíbvgš¿YvjqGescÖevmxKj¨vYI•e‡`wkKKg©ms¯’vb gš¿Yvj‡qimn:
cwiPvjK:06
12wU 9wU
11wU 10wU DËi: K
12 wU: 44 22 3 22 3 21  21 †_‡K 70 ch©šÍ
11wU I 1 wU 71 = 11 + 1 = 12 wU|
163. 25 †_‡K 55-Gi g‡a¨ †gŠwjK msL¨v Av‡Q? Dc‡RjvI
_vbvwkÿvAwdmvi:05
4wU 6 wU
7 wU 9 wU DËi: M
7wU: 29, 31, 37, 41, 43, 47 Ges 53|
164. 43 †_‡K 60-Gi g‡a¨ †gŠwjK msL¨v - 26Zg wewmGm
5 3
7 4 DËi: N
4wU: 43, 47, 53, 59|
165. 45 †_‡K 72 -Gi g‡a¨ KqwU †gŠwjK Av‡Q?
Sonali, Janata and Agrani Bank senior officer: 08
5 6
7 8 DËi: L
6wU: 47, 53, 59, 61, 67 Ges 71|
166. 56 †_‡K 100 Gi g‡a¨ †gŠwjK msL¨v KqwU? EXIM
BankLtd. (TraineeAsst.Officer)2018
8 9
10 11 DËi: L
167. 50 Gi †P‡q †QvU KZwU †gŠwjK msL¨v Av‡Q? Janata
BankLtd. (Asst.Officer)2015
14 15
16 18 DËi: L
168. 50 †_‡K 103 ch©šÍ KZwU †gŠwjK msL¨v Av‡Q?
cÖwZiÿvgš¿Yvj‡qiAaxbGWwgwb‡÷ªkbAwdmviIcv‡m©vbvjAwdmvi:06
10wU 11wU
12wU 13wU DËi: M
169. 90 †_‡K 100 Gi g‡a¨ KqwU †gŠwjK msL¨v Av‡Q?
kÖgIKg©ms¯’vbgš¿Yvj‡qiAaxbKjKviLvbvIcÖwZôvb cwi`k©bcwi`ß‡ii
mnKvix cwi`k©K:05
2wU 1wU
3wU GKwUI bq DËi: L
170. 100 †_‡K 110 ch©šÍ msL¨v¸‡jvi g‡a¨ KqwU †gŠwjK
msL¨v i‡q‡Q? evsjv‡`k†ijI‡qnvmcvZvjmg~nmn:mvR©b:05;
PviwU GKwU
`yBwU wZbwU DËi: K
171. 100 -Gi ‡P‡q eo Ges 150-Gi †P‡q †QvU KqwU
†gŠwjK msL¨v Av‡Q? ivóªvqËe¨vsKAwdmvi:97
7wU 8wU
9wU 10wU DËi: N
(N) x I y msL¨vi g‡a¨ e„nËg I ÿz`ªZg †gŠwjK msL¨vi AšÍi wbY©q I Ab¨vb¨
172. 60 †_‡K 80 -Gi gaëZx© e„nËg I ÿz`ªZg †gŠwjK
msL¨vi AšÍi n‡e- 27 Zg wewmGm
8 12
18 140 DËi: M
60 I 80 Gi gv‡S †gŠwjK msL¨vmg~n : 61, 67,
71, 73, 79| G‡ì gv‡S ÿz`ªZg †gŠwjK msL¨v 61
I e„nËg †gŠwjK msL¨v 79|
 G‡ì cv_©K¨ = 79 - 61 = 18|
173. 30 †_‡K 80 Gi gaëZ©x e„nËg I ÿz`ªZg †gŠwjK
msL¨vi eëavb KZ? RajshahiKrishiUnnayanBank
(cashier) :17;mgevq Awa. wØZxq †kÖYxi †M‡R‡UW Awdmvi: 97
35 42
48 55 DËi: M
cv_©K¨ = 79 - 31 = 48|
174. 30 †_‡K 90 Gi gaëZx© e„nËg I ÿz`ªZg †gŠwjK
msL¨vi AšÍi KZ? _vbv I †Rjv mgvR‡mev Awdmvi:99
58 42
68 62 DËi: K
30 31(†gŠwjK)... (†gŠwjK)89 90| myZivs
†gŠwjK msL¨v `ywUi AšÍi = 89-31 = 58|
175. 40 †_‡K 100 ch©šÍ e„nËg I ÿz`ªZg †gŠwjK
msL¨vi AšÍi KZ? Lv`Äwa`ß‡iiAax‡bLv`¨cwi`k©K:00
59 56
60 70 DËi: L
40 41(†gŠwjK msL¨v) ..... (†gŠwjK msL¨v) 97
 100|  cv_©K¨ = 97-41 = 56|
mgvavb
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Math Tutor 31
176. 10 †_‡K 60 ch©šÍ †h mKj †gŠwjK msL¨vi GKK
¯’vbxq A¼ 9 Zv‡`i mgwó KZ? RajshahiKrishi
UnnayanBank (Supervisor):17
146 99
105 107 DËi: N
177. cÖ_g 9wU †gŠwjK msL¨vi Mo- BangladeshBank Asst.
Director:14
9 11
11
9
1
11
9
2
DËi: M
cÖ_g 9 wU †gŠwjK msL¨vi †hvMdj =
2+3+5+7+11+13+17+19+23 = 100
 Mo =
9
100
= 11
9
1
|
178. wb‡Pi †KvbwU cÖ_g 5wU †gŠwjK msL¨vi Mo?
BangladeshHouseBuildingFinanceCorporation(so):17
4.5 5.6
7.5 8.6 DËi: L
179. 30 †_‡K 50 Gi g‡a¨ mKj †gŠwjK msL¨vi Mo
KZ? AgraniBankLtd.SeniorOfficer:17(Cancelled)
37 37.8
39.8 39 DËi: M
30 †_‡K 50 Gi gv‡Si †gŠwjK msL¨vi †hvMdj
= 31+37+41+43+47 = 199
 Mo =
5
199
= 39
5
4
= 39.8|
01.16 †Rvo msL¨v I we‡Rvo msL¨v
 †R‡b wbb – 18 ( †Rvo I we‡Rvo msL¨v msµvšÍ mgvav‡bi †KŠkj)
 µwgK †Rvo I we‡Rvo m¤ú‡K© †ewmK Av‡jvPbv Ô‡R‡b wbb-00Õ †_‡K c‡o wbb|
 µwgK †e‡Rvo/AhyM¥/ abvZ¥K we‡Rvo : cÖ‡kœ µwgK we‡Rvo/AhyM¥/abvZ¥K we‡Rvo _vK‡j 1 ewm‡q mgvavb Kiæb|
 µwgK FYvZ¥K we‡Rvo: cÖ‡kœ µwgK FYvZ¥K we‡Rvo _vK‡j -1 ewm‡q mgvavb Kiæb|
 µwgK †Rvo/hyM¥: cÖ‡kœ µwgK †Rvo/hyM¥/abvZ¥K †Rvo _vK‡j 2 ewm‡q mgvavb Kiæb|
 µwgK FYvZ¥K †Rvo: cÖ‡kœ µwgK FYvZ¥K †Rvo _vK‡j -2 ewm‡q mgvavb Kiæb|
 abvZ¥K c~Y©msL¨v: 1, 2, 3, 4, 5, 6, 7, 8, 9 BZ¨vw`|
 FYvZ¥K c~Y©msL¨v: -1, -2, -3, -4, -5, -6, -7, -8, -9 BZ¨vw`|
 GKvwaK we‡Rvo msL¨vi ¸Ydj me mgq we‡Rvo nq| †hgb- 357 = 105|
180. x I y DfqB we‡Rvo msL¨v n‡j †Rvo msL¨v
n‡e? 32Zg wewmGm(we‡kl)
x+y+1 xy
xy + 2 x+y DËi: N
x = 1 I y = 1| †Rvo msL¨v = ? 1+1+1
= 3 (mwVK bq) 11 = 1 (mwVK bh) 1
 1 + 2 = 3 (mwVK bq) 1 + 1 = 2 (mwVK)
181. hw` x GKwU FYvZ¥K we‡Rvo c~Y©msL¨v nq Ges
y GKwU abvZ¥K †Rvo msL¨v nq, Z‡e xy
Aek¨B- GgweGg: 06
†Rvo Ges FYvZ¥K we‡Rvo Ges FYvZ¥K
†Rvo Ges abvZ¥K we‡Rvo Ges abvZ¥K
x = - 1, y = 2. xy = - 12 = - 2|
xy Aek¨B †Rvo I FYvZ¥K n‡e | DËi: K
182. hw` 𝒂 & 𝑏 DfqB abvZ¥K †Rvo c~Y©msL¨v nq, Z‡e
wb‡Pi †KvbwU Aek¨B †Rvo msL¨v n‡e? MBA : 06
1) ab
2) (a + 1)b
3) ab+1
1 only 1& 2
1 & 3 1, 2 & 3 DËi: M
a = 2, b = 2. †Rvo msL¨vi Ackb Lyu‡R †ei
Ki‡Z n‡e| 1) ab
= 22
= 4 (†Rvo)
2) (a + 1)b
= (2+1)2
= 9 (we‡Rvo)
3) ab+1
= 22+1
= 8 (†Rvo)
1 I 3 bs-G †Rvo msL¨v G‡m‡Q, hv Ackb
†Z Av‡Q|
183. wb‡Pi †KvbwU Aek¨B we‡Rvo msL¨v n‡e? BGgweG
(Xvwe): GwcÖj -07
1) `yBwU †Rvo msL¨vi ¸Ydj
2) `yBwU we‡Rvo msL¨vi ¸Ydj
3) GKwU †Rvo Ges GKwU we‡Rvo msL¨vi †hvMdj
1, 2 & 3 1 only
2 & 3 only 1 & 3 only DËi:
†Rvo = 2, we‡Rvo = 1| cÖkœvbyhvqx we‡Rvo msL¨v
†ei Ki‡Z n‡e|
mgvavb
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32Math Tutor
1) 22= 4 (†Rvo) 2) 11 = 1 (we‡Rvo)
3) 2+ 1= 3 (we‡Rvo)
2 I 3 bs-G we‡Rvo msL¨v G‡m‡Q, hv Ackb †Z
Av‡Q|
184. hw` n Ges p `ywU AhyM¥ msL¨v nq, Z‡e wb‡¤œi
†KvbwU Aek¨ hyM¥ msL¨v n‡e? c~evjx e¨vsKt 06/ _vbv
wkÿv Awdmvit 99
n+p np
np+2 n+p+1 DËi: K
185. hw` n Ges p `ywU †Rvo msL¨v nq, Z‡e wb‡¤œi
†KvbwU Aek¨B we‡Rvo msL¨v n‡e? AMÖYxe¨vsKAwdmvi:08
n+2p np+1
n + p 2n+p DËi: L
186. hw` m GKwU †Rvo c~Y©msL¨v Ges n GKwU we‡Rvo
c~Y©msL¨v nq Ges Dfq msL¨vB abvZ¥K nq, Z‡e
wb‡Pi †KvbwU Aek¨B abvZ¥K †Rvo msL¨v n‡e?
IBA(MBA):87-88
m2
+n2
mn + n2
m3
+n3
mn+𝑚2
DËi: N
187. hw` x GKwU abvZ¥K †Rvo msL¨v nq, Z‡e wb‡Pi
†KvbwU e¨ZxZ Ab¨ mKj DËi we‡Rvo n‡e? IBA
(MBA):05-06
(x+3) (x+5) x2
+ 5
x2
+ 6x +9 3x2
+ 4 DËi: N
188. hw` m I n `ywU FYvZ¡K c~Y©msL¨v nq, Z‡e wb‡¤œi
†KvbwU Aek¨B mwVK? DutchBanglaBankLtd. :17
m + n < 0 m – n < 0
mn < 0 None DËi: K
†h‡Kvb `ywU FYvZ¥K c~Y©msL¨v a‡i †bqv hvK:
m = -2 I n = -3
m + n < 0 ev, (-2) + (-3) < 0 = -5 < 0
GLv‡b -5, 0 Gi †P‡q †QvU, ZvB GwUB mwVK|
189. hw` 2x – 3 we‡Rvo msL¨v nq Z‡e cieZ©x †Rvo
msL¨v †ei Kiæb|FirstSecurityIslamiBnakLtd.Officer:14
2x - 5 2x - 4
2x - 2 4x + 1 DËi: M
we‡Rvo Gi mv‡_ 1 †hvM Ki‡j cieZx© †Rvo
msL¨v cvIqv hvq| Avevi †Rv‡oi mv‡_ 1 †hvM Ki‡j
cieZx© we‡Rvo msL¨v cvIqv hvq| cÖ‡kœ 2x – 3 n‡”Q
GKwU we‡Rvo msL¨v, Gi cieZx© †Rvo msL¨v †ei
Kivi Rb¨ 1 †hvM Ki‡Z n‡e- 2x – 3 + 1
= 2x - 2 |
190. hw` 3x+1GKwU we‡Rvo msL¨v wb‡`©k K‡i, Z‡e
wb‡Pi †KvbwU Zvi cieZx© we‡Rvo msL¨v n‡e?
MBA : 05
3(x+1) 3(x+2)
3(x+3) 3x+2 DËi: K
GKwU we‡Rvo msL¨v †_‡K cieZx© we‡Rvo msL¨v
†ei Ki‡Z n‡j 2 †hvM Ki‡Z nq| cÖ‡kœ cÖ`Ë 3x+1
n‡”Q GKwU we‡Rvo msL¨v| cieZx© we‡Rvo msL¨v
†ei Ki‡Z n‡j 3x+1 Gi mv‡_ 2 †hvM Ki‡Z n‡e
A_©vr, (3x+1) + 2 = 3x + 1 + 2 = 3x + 3 =
3 (3 x+ 1) |
191. hw` n – 5 GKwU †Rvo c~Y©msL¨v nq, Z‡e cieZ©x
†Rvo µwgK c~Y©msL¨v †KvbwU? BangladeshHouse
BuildingFinanceCorporation (SO):17
n - 7 n - 3
n - 4 n - 2 DËi: L
GKwU †Rvo msL¨v †_‡K cieZx© †Rvo msL¨v †ei
Ki‡Z n‡j 2 †hvM Ki‡Z nq| cieZx© †Rvo msL¨v
†ei Ki‡Z n‡j n – 5 Gi mv‡_ 2 †hvM Ki‡Z n‡e
A_©vr, n – 5 + 2 = n – 3 |
192. cvuPwU c~Y© msL¨vi ¸Ydj hw` we‡Rvo msL¨v nq
Zvn‡j D³ cvuPwU c~Y©msL¨vi wVK KqwU we‡Rvo
n‡e? kÖ: cwi: 05
2 3
4 5 DËi: 5
GKvwaK msL¨vi ¸Ydj †e‡Rvo n‡Z n‡j GKvwaK
msL¨vi cÖwZwUB †e‡Rvo n‡Z n‡e, ZvB GLv‡b D³
cvuPwU c~Y© msL¨vi me KqwU we‡Rvo|
01.17 g~j` I Ag~j` msL¨v
 †R‡b wbb – 19
 g~j` msL¨v(Rational Number)t
 k~Y¨ I mKj ¯^vfvweK msL¨v g~j` msL¨v| †hgb: 0, 1, 2, 3 BZ¨vw` |
 cÖK…Z I AcÖK…Z mKj fMœvsk g~j` msL¨v | †hgb:
2
1
,
5
11
8
7
, BZ¨vw`|
mgvavb
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mgvavb
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mgvavb
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mgvavb
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Math Tutor 33
 `kwg‡Ki c‡ii Ni¸‡jv mmxg n‡j msL¨vwU g~j` msL¨v| †hgb: 4. 678 |
 mKj c~Y© eM© ¯^vfvweK msL¨vi eM©g~j g~j` msL¨v| †hgb- 49 = 7, 64 = 8, 121 = 11 BZ¨vw`|
 mKj c~Y© Nb ¯^vfvweK msL¨vi Nbg~j g~j` msL¨v| †hgb- 3
27 = 3, 3
125 = 5 BZ¨vw`|
 `kwg‡Ki c‡ii Ni¸‡jv †cŠY‡cŠwYK AvKv‡i Amxg n‡j| †hgb:
3
4
= 1.33333... = 1. 3
 ,
3
10
=
3.3333.., = 3. 3
 BZ¨vw`|
 Ag~j` msL¨v(Irrational Number)t
 `kwg‡Ki c‡ii Ni¸‡jv hw` wfbœ wfbœ AvKv‡i Amxg nq, Zvn‡j msL¨vwU Ag~j` msL¨v| †hgb: 3.142857...
 mKj †gŠwjK msL¨v, c~Y©eM© I c~Y©Nb bq Ggb mKj msL¨vi eM©g~j Ges Nbg~j me mgq Ag~j` msL¨v|
†hgb: 3
3
11
5
3
2 ,
,
, , 12 , 3
22 BZ¨vw`|
 K‡qKwU weL¨vZ Ag~j` msL¨vt
 cvB t 𝜋 GKwU Ag~j` msL¨v| GLv‡b, 𝜋 = 3.14285... |
 Aqjvi msL¨v t e GKwU Ag~j` msL¨v| GLv‡b, e = 2.71828....|
 dvB (†mvbvjx AbycvZ) t 𝜑 n‡”Q GKwU Ag~j` msL¨v| GLv‡b, 𝜑 = 1.618033...
193. hw` p GKwU †gŠwjK msL¨v nq Z‡e P - 26Zg
wewmGm
GKwU ¯^vfvweK msL¨v GKwU c~Y©msL¨v
GKwU g~j` msL¨v GKwU Ag~j` msL¨v DËi:N
194. 2 msL¨vwU wK msL¨v ? 25Zg wewmGm
GKwU ¯^vfvweK msL¨v GKwU c~Y© msL¨v
195. 5 wK ai‡bi msL¨v?AvbmviIwfwWwcAwa:mv‡K©jA¨vWRyU¨v:05
GKwU ¯^vfvweK msL¨v GKwU c~Y© msL¨v
196. 7 3 msL¨v †Kvb ai‡bi msL¨v? 12Zg wbeÜb
RwUj msL¨v g~j` msL¨v
Ag~j` msL¨v ev¯Íe msL¨ DËi: M
197. wb‡Pi †KvbwU g~j` msL¨v? 9gwkÿKwbeÜb:13
2 3
8
3
9 2
8 DËi: L
8 n‡”Q c~Y© Nb msL¨v ZvB 3
8 n‡”Q g~j`
msL¨v| 3
8 = 3 3
2 = 2 (GKwU g~j` msL¨v)
198. †h msL¨v‡K `ywU c~Y© msL¨vi fvMdj AvKv‡i cÖKvk
Kiv hvq bv Zv‡K wK e‡j? cÖavbgš¿xiKvh©vj‡qimnKvixcwiPvjK,
M‡elYvKg©KZ©v,†Uwj‡dvbBwÄwbqviImnKvixKvw¤úDUvi†cÖvMÖvgvi:13
g~j` msL¨v ¯^vfvweK msL¨v
Ag~j` msL¨v RwUj msL¨v DËi: M
199. g~j` msL¨vi †mU †evSvq wb‡Pi †KvbwU‡K? cÖevmxKj¨vY
I •e‡`wkKKg©ms¯’vbgš¿Yvj‡qimnKvixcwiPvjK:12
Z Q
P N DËi: L
Z n‡”Q c~Y© msL¨vi †mU, Q n‡”Q g~j` msL¨vi
†mU, P n‡”Q †gŠwjK msL¨vi †mU Ges N n‡”Q
¯^vfvweK msL¨vi †mU|
200. wb‡Pi †KvbwU Ag~j` msL¨v? lôcÖfvlKwbeÜbIcÖZ¨qb:10
9
16
2
4
49
26
64
DËi: L I N
201. wb‡Pi †KvbwU g~j` msL¨v? WvK,†Uwj‡hvMv‡hvMIZ_¨cÖhyw³
gš¿Yvj‡qimnKvix†cÖvMÖvgvi:17
243
3
343
3
392
3
676
3
DËi: L
202. wb‡Pi †KvbwU Ag~j` msL¨v? wewfbœ gš¿YvjqmnKvix†gBb‡Ub¨vÝ
BwÄwbqcvi:17
27
3
125
3
5
81
4
4
32
5
8
DËi: K
203. wb‡Pi †KvbwU Ag~j` msL¨v? 18Zg†emiKvixwkÿKwbeÜb(¯‹zj
mgch©vq):17
𝜋 2
11 me¸‡jv DËi: N
N
M
L
K
N
M
L
K
N
M
L
K
N
M
L
K
mgvavb
N
M
L
K
N
M
L
K
mgvavb
N
M
L
K
N
M
L
K
N
M
L
K
N
M
L
K
N
M
L
K

34Math Tutor
01.18 wewmGm wjwLZ cÖkœ mgvavb
204. `yB A¼wewkó GKwU msL¨v‡K A¼Ø‡qi ¸Ydj Øviv
fvM Ki‡j fvMdj 3 nq| H msL¨vwUi mv‡_ 18 †hvM
Ki‡j A¼Øq ¯’vb wewbgq K‡i| msL¨vwU KZ? 34Zg
wewmGm wjwLZ
g‡bKwi, GKK ¯’vbxq A¼ = x Ges `kK ¯’vbxq
A¼ = y
 msL¨vwU = 10y + x
A¼Øq ¯’vb wewbgq Ki‡j = 10x + y
1g kZ©vbymv‡i,
xy
x
y
10 
= 3
ev, 10y + x = 3xy ………(i)
2q kZ©vbymv‡i, 10y + x + 18 = 10x + y
ev, 9x = 9y + 18
ev, 9x - 9y = 18
ev, 9 (x - y) = 18
ev, x - y = 2
ev, x = 2 + y ………………(ii)
(i) bs mgxKi‡Y x = 2 + y ewm‡q cvB,
10y + 2 + y = 3(2+y)y
ev, 11y + 2 = 6y + 3y2
ev, 3y2
+ 6y - 11y -2 = 0
ev, 3y2
- 5y - 2 = 0
ev, 3y2
- 6y + y - 2 = 0
ev, 3y (y-2) + 1(y-2) = 0
ev, (y-2) (3y + 1) = 0
 y = 2 A_ev y = -
3
1
FYvZ¥K gvb MÖnY‡hvM¨ bv nIqvq y = 2 n‡e|
(ii) bs mgxKi‡Y y = 2 ewm‡q cvB,
x = 2 + 2 = 4
myZivs, wb‡Y©q msL¨vwU = 10y + x = 102 + 4
= 24. (DËi)
205. `yB A¼wewkó †Kvb msL¨vi A¼Ø‡qi mgwó 9| A¼Øq
¯’vb wewbgq Ki‡j †h msL¨v cvIqv hvq Zv cÖ`Ë
msL¨v n‡Z 45 Kg| msL¨vwU wbY©q Kiæb| 31Zg
wewmGm wjwLZ
g‡bKwi, GKK ¯’vbxq A¼ x n‡j `kK ¯’vbxq A¼
(9 - x)
msL¨vwU = 10(9-x) + x = 90 - 9x
¯’vb wewbgq Ki‡j msL¨vwU = 10x + (9 -x)
= 10x + 9 -x
= 9x + 9
kZ©g‡Z, 9x + 9 + 45 = 90 - 9x
ev, 9x + 9x = 90 - 54
ev, 18x = 36
 x =
18
36
= 2
myZivs, msL¨vwU = 90 - 92 = 90 - 18 = 72.
206. `yB A¼wewkó †Kvb msL¨vi `kK ¯’vbxq A¼wU GKK
¯’vbxq A¼ n‡Z 5 eo| msL¨vwU †_‡K A¼Ø‡qi
mgwói cuvP¸Y we‡qvM Ki‡j A¼Ø‡qi ¯’vb wewbgq
nq| msL¨vwU KZ? 23Zg wewmGm wjwLZ
g‡bKwi, GKK ¯’vbxq A¼ x
Ges `kK ¯’vbxq A¼ x + 5
msL¨vwU = 10 (x+5) + x
kZ©g‡Z, 10 (x+5) + x-5(x+5+x) = 10x + x + 5
ev, 10x + 50 + x-5x -25-5x = 11x +5
ev, x+25 = 11x + 5
ev, 10x = 20
 x = 2
myZivs, wb‡Y©q msL¨vwU= 10 (x+5) + x
= 10 (2+5) + 2
= 70 + 2
= 72 (DËi)
01.19 Ab¨vb¨ wjwLZ cÖkœ mgvavb
207. `yB A¼wewkó GKwU msL¨vi GKK ¯’vbxq A¼ `kK
¯’vbxq A‡¼i wZb¸Y A‡cÿv GK †ewk| A¼Øq ¯’vb
wewbgq Ki‡j †h msL¨v cvIqv hvq, Zv A¼Ø‡qi
mgwói AvU¸‡Yi mgvb| msL¨vwU KZ? cÖwZiÿvgš¿Yvj‡qi
AwdmmnKvixKvg-Kw¤úDUvigy`ªvÿwiK2019; beg`kg†kÖwYiMwYZ:
Abykxjbx 12.4Gi12bscÖkœ
g‡bKwi, `kK ¯’vbxq A¼ = x
Ges GKK ¯’vbxq A¼ = 3x +1
 msL¨vwU = x10 + 3x + 1
= 10x + 3x + 1 = 13x + 1
A¼Øq ¯’vb wewbgq Ki‡j = x + (3x +1) 10
mgvavb
mgvavb
mgvavb
mgvavb
ïay wjwLZ Av‡jvPbv
 

Math Tutor 35
= x + 30x + 10
= 31x + 10
cÖkœg‡Z, 31x + 10 = (x+3x+1) 8
ev, 31x + 10 = 8x + 24x + 8
ev, 10 - 8 = 32x - 31x
ev, x = 2
 msL¨vwU = 132 + 1 = 26 + 1 = 27 (DËi)
208. `ywU msL¨v Ggb †h, cÖ_g msL¨v wØZxq msL¨v †_‡K
30 MÖnY Ki‡j Zv‡`i AbycvZ 2 : 1 nq| wKš‘ hw`
wØZxq msL¨v cÖ_g msL¨v †_‡K 50 MÖnY K‡i Z‡e
Zv‡`i AbycvZ nq 1 : 3| msL¨v `ywU KZ? evsjv‡`k
†ijI‡qieywKsmnKvix2029
g‡b Kwi, cÖ_g msL¨v = x Ges wØZxq msL¨v = y
1g kZ©vbymv‡i,
1
2
30
y
30
x



ev, x + 30 = 2y - 60
ev, x = 2y - 90 ……… (i)
2q kZ©vbymv‡i,
3
1
50
y
50
x



ev, 3x - 150 = y + 50
ev, 3x - y = 200 …….. (ii)
(ii) bs mgxKi‡Y x = 2y - 90 ewm‡q cvB,
3 (2y -90) - y = 200
ev, 6y - 270 - y = 200
ev, 5y = 200 + 270
ev, 5y = 470
 y =
5
470
= 94
y Gi gvb (ii) bs mgxKi‡Y ewm‡q cvB,
x = 294 - 90 = 188 - 90 = 98
AZGe, cÖ_g msL¨vwU 98 Ges
wØZxq msL¨vwU 94 (DËi)
01.20 cvV¨ eB †_‡K wjwLZ mgm¨v mgvavb
209. `yB A¼ wewkó GKwU msL¨vi A¼Ø‡qi AšÍi 4;
msL¨vwUi A¼Øq ¯’vb wewbgq Ki‡j †h msL¨v cvIqv
hvq, Zvi I g~j msL¨vwUi †hvMdj 110; msL¨vwU
wbY©q Ki| ; beg`kg†kÖwYiMwYZ:Abykxjbx 12.4Gi12bscÖkœ
g‡b Kwi, GKK ¯’vbxq A¼ x
Ges `kK ¯’vbxq A¼ y.
 msL¨vwU = x + 10y
1g kZ©vbymv‡i, x - y = 4 ……. (i) [ hLb, x>y]
Avevi, y- x = 4 …… (ii) [hLb, y>x]
A¼Øq ¯’vb wewbgq Ki‡j msL¨vwU n‡e = 10x + y
2q kZv©bymv‡i, 10x + y + x + 10y = 4 + 10
ev, 11x + 11y = 110
ev, 11 (x + y) = 110
 x + y = 10 ………... (iii)
(i) I (iii) bs mgxKiY †hvM K‡i cvB,
x - y = 4
x + y = 10
2x = 14
 x = 7
Avevi, (iii) bs †_‡K (ii) bs we‡qvM K‡i cvB,
y - x - x - y = 4 -10
ev, -2x = -6
 x = 3
(iii) bs mgxKi‡Y x = 7 ewm‡q cvB,
7 + y = 10
 y = 10 - 7 = 3.
Avevi, (iii) bs mgxKi‡Y x = 3 ewm‡q cvB,
3 + y = 10
 y = 10 - 3 = 7.
AZGe, x= 7 Ges y = 3 n‡j,
msL¨vwU = x + 10y = 7 + 103 = 37
A_ev, x = 3 Ges y = 7 n‡j.
msL¨vwU = = x + 10y = 3 + 107 = 73.
DËi: wb‡Y©q msL¨vwU 37 A_ev 73 |
mgvavb
mgvavb

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